In

$\backslash det\backslash left(I\backslash right)\; =\; 1$, where $I$ is an The determinant is '' multilinear'': if the ''j''th column of a matrix $A$ is written as a The determinant is '' alternating'': whenever two columns of a matrix are identical, its determinant is 0:
#: $,\; a\_1,\; \backslash dots,\; v,\; \backslash dots,\; v,\; \backslash dots,\; a\_n,\; =\; 0.$
If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any $n\; \backslash times\; n$-matrix ''A'' a number that satisfies these three properties. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.
To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a

_{''m''} and ''I''_{''n''} are the and identity matrices, respectively.
From this general result several consequences follow.

_{''l''} = −(''l'' – 1)! tr(''A''^{''l''}) as
:$\backslash det(A)\; =\; \backslash frac\; B\_n(s\_1,\; s\_2,\; \backslash ldots,\; s\_n).$
This formula can also be used to find the determinant of a matrix with multidimensional indices and . The product and trace of such matrices are defined in a natural way as
:$(AB)^I\_J\; =\; \backslash sum\_K\; A^I\_K\; B^K\_J,\; \backslash operatorname(A)\; =\; \backslash sum\_I\; A^I\_I.$
An important arbitrary dimension identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of ''A'' is less than 1 in absolute value,
:$\backslash det(I\; +\; A)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash left(-\backslash sum\_^\backslash infty\; \backslash frac\; \backslash operatorname\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$
where is the identity matrix. More generally, if
:$\backslash sum\_^\backslash infty\; \backslash frac\; \backslash left(-\backslash sum\_^\backslash infty\; \backslash frac\backslash operatorname\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$
is expanded as a formal power series in then all coefficients of ^{} for are zero and the remaining polynomial is .

^{''n''}, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in R^{''n''} represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation (vector space), orientation of the basis is consistent with or opposite to the orientation of the

^{''n''} (the domain of ''f''), the integral over ''f''(''U'') of some other function is given by
:$\backslash int\_\; \backslash phi(\backslash mathbf)\backslash ,\; d\backslash mathbf\; =\; \backslash int\_U\; \backslash phi(f(\backslash mathbf))\; \backslash left,\; \backslash det(\backslash operatornamef)(\backslash mathbf)\backslash \; \backslash ,d\backslash mathbf.$
The Jacobian also occurs in the inverse function theorem.

Determinant Interactive Program and Tutorial

Linear algebra: determinants.

Compute determinants of matrices up to order 6 using Laplace expansion you choose.

Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.

{{authority control Determinants, Matrix theory Linear algebra Homogeneous polynomials Algebra

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the determinant is a scalar value that is a function
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of the entries of a square matrix
In mathematics
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. It allows characterizing some properties of the matrix and the linear map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix is denoted , , or .
In the case of a matrix the determinant can be defined as
:$\backslash begin,\; A,\; =\; \backslash begin\; a\; \&\; b\backslash \backslash c\; \&\; d\; \backslash end=ad-bc.\backslash end$
Similarly, for a 3 × 3 matrix ''A'', its determinant is
:$\backslash begin\; ,\; A,\; =\; \backslash begin\; a\; \&\; b\; \&\; c\; \backslash \backslash \; d\; \&\; e\; \&\; f\; \backslash \backslash \; g\; \&\; h\; \&\; i\; \backslash end\; \&=\; a\backslash ,\backslash begin\; e\; \&\; f\; \backslash \backslash \; h\; \&\; i\; \backslash end\; -\; b\backslash ,\backslash begin\; d\; \&\; f\; \backslash \backslash \; g\; \&\; i\; \backslash end\; +\; c\backslash ,\backslash begin\; d\; \&\; e\; \backslash \backslash \; g\; \&\; h\; \backslash end\; \backslash \backslash $ &= aei + bfg + cdh - ceg - bdi - afh.
\end
Each determinant of a matrix in this equation is called a minor
Minor may refer to:
* Minor (law), a person under the age of certain legal activities.
** A person who has not reached the age of majority
* Academic minor, a secondary field of study in undergraduate education
Music theory
*Minor chord
** Barbe ...

of the matrix . This procedure can be extended to give a recursive definition for the determinant of an matrix, known as ''Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . S ...

''.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficient
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s in a system of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and determinants can be used to solve these equations (Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...

), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

of a matrix, whose roots are the eigenvalue
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

s. In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, the signed -dimensional volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
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of a -dimensional parallelepiped
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

is expressed by a determinant. This is used in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

with exterior differential form
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

s and the Jacobian determinant
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

, in particular for changes of variables in multiple integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
2 × 2 matrices

The determinant of a matrix $\backslash begin\; a\; \&\; b\; \backslash \backslash c\; \&\; d\; \backslash end$ is denoted either by "" or by vertical bars around the matrix, and is defined as :$\backslash det\; \backslash begin\; a\; \&\; b\; \backslash \backslash c\; \&\; d\; \backslash end\; =\; \backslash begin\; a\; \&\; b\; \backslash \backslash c\; \&\; d\; \backslash end\; =\; ad\; -\; bc.$ For example, :$\backslash det\; \backslash begin\; 3\; \&\; 7\; \backslash \backslash 1\; \&\; -4\; \backslash end\; =\; \backslash begin\; 3\; \&\; 7\; \backslash \backslash \; 1\; \&\; \backslash end\; =\; 3\; \backslash cdot\; (-4)\; -\; 7\; \backslash cdot\; 1\; =\; -19.$First properties

The determinant has several key properties that can be proved by direct evaluation of the definition for $2\; \backslash times\; 2$-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of theidentity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

$\backslash begin1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end$ is 1.
Second, the determinant is zero if two rows are the same:
:$\backslash begin\; a\; \&\; b\; \backslash \backslash \; a\; \&\; b\; \backslash end\; =\; ab\; -\; ba\; =\; 0.$
This holds similarly if the two columns are the same. Moreover,
:$\backslash begina\; \&\; b\; +\; b\text{'}\; \backslash \backslash \; c\; \&\; d\; +\; d\text{'}\; \backslash end\; =\; a(d+d\text{'})-(b+b\text{'})c\; =\; \backslash begina\; \&\; b\backslash \backslash \; c\; \&\; d\; \backslash end\; +\; \backslash begina\; \&\; b\text{'}\; \backslash \backslash \; c\; \&\; d\text{'}\; \backslash end.$
Finally, if any column is multiplied by some number $r$ (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
:$\backslash begin\; r\; \backslash cdot\; a\; \&\; b\; \backslash \backslash \; r\; \backslash cdot\; c\; \&\; d\; \backslash end\; =\; rad\; -\; brc\; =\; r(ad-bc)\; =\; r\; \backslash cdot\; \backslash begin\; a\; \&\; b\; \backslash \backslash c\; \&\; d\; \backslash end.$
Geometric meaning

If the matrix entries are real numbers, the matrix can be used to represent twolinear map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s: one that maps the standard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

vectors to the rows of , and one that maps them to the columns of . In either case, the images of the basis vectors form a parallelogram
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

that represents the image of the unit square
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at , , , and , as shown in the accompanying diagram.
The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by . (The parallelogram formed by the columns of is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)
The absolute value of the determinant together with the sign becomes the ''oriented area'' of the parallelogram. The oriented area is the same as the usual area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

).
To show that is the signed area, one may consider a matrix containing two vectors and representing the parallelogram's sides. The signed area can be expressed as for the angle ''θ'' between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

this already is the signed area, yet it may be expressed more conveniently using the cosine
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

of the complementary angle to a perpendicular vector, e.g. , so that , which can be determined by the pattern of the scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

to be equal to :
: $\backslash text\; =\; ,\; \backslash boldsymbol,\; \backslash ,,\; \backslash boldsymbol,\; \backslash ,\backslash sin\backslash ,\backslash theta\; =\; \backslash left,\; \backslash boldsymbol^\backslash perp\backslash \backslash ,\backslash left,\; \backslash boldsymbol\backslash \backslash ,\backslash cos\backslash ,\backslash theta\text{'}\; =\; \backslash begin\; -b\; \backslash \backslash \; a\; \backslash end\; \backslash cdot\; \backslash begin\; c\; \backslash \backslash \; d\; \backslash end\; =\; ad\; -\; bc.$
Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by ''A''. When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving.
The object known as the ''bivector In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

'' is related to these ideas. In 2D, it can be interpreted as an ''oriented plane segment'' formed by imagining two vectors each with origin , and coordinates and . The bivector magnitude (denoted by ) is the ''signed area'', which is also the determinant .
If an real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

matrix ''A'' is written in terms of its column vectors $A\; =\; \backslash left;\; href="/html/ALL/s/begin\_\backslash mathbf\_1\_\_\backslash mathbf\_2\_\_\backslash cdots\_\_\backslash mathbf\_n\backslash end\backslash right.html"\; ;"title="begin\; \backslash mathbf\_1\; \backslash mathbf\_2\; \backslash cdots\; \backslash mathbf\_n\backslash end\backslash right">begin\; \backslash mathbf\_1\; \backslash mathbf\_2\; \backslash cdots\; \backslash mathbf\_n\backslash end\backslash right$ to the ''n''-dimensional parallelotope defined by the vectors $\backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash ldots,\; \backslash mathbf\_n,$ the region $P\; =\; \backslash left\backslash .$
The determinant gives the signed
Signing may refer to:
* Using sign language
* Signature, placing one's name on a document
* Signature (disambiguation)
* Manual communication, signing as a form of communication using the hands in place of the voice
* Digital signature, signing as ...

''n''-dimensional volume of this parallelotope, $\backslash det(A)\; =\; \backslash pm\; \backslash text(P),$ and hence describes more generally the ''n''-dimensional volume scaling factor of the linear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

produced by ''A''. (The sign shows whether the transformation preserves or reverses orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building design ...

.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully ''n''-dimensional, which indicates that the dimension of the image of ''A'' is less than ''n''. This means
Means may refer to:
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History
The band was formed in 2001 under the name of Means 2 An End as ...

that ''A'' produces a linear transformation which is neither onto
In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

nor one-to-one
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual commun ...

, and so is not invertible.
Definition

In the sequel, ''A'' is asquare matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with ''n'' rows and ''n'' columns, so that it can be written as
:$A\; =\; \backslash begin\; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash end.$
The entries $a\_$ etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are elements in more abstract algebraic structures known as commutative ring
In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative.
Definition and first e ...

s.
The determinant of ''A'' is denoted by det(''A''), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
:$\backslash begin\; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash end.$
There are various equivalent ways to define the determinant of a square matrix ''A'', i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.
Leibniz formula

The ''Leibniz formula'' for the determinant of a matrix is the following: :$\backslash begin\; \backslash begina\&b\&c\backslash \backslash d\&e\&f\backslash \backslash g\&h\&i\backslash end\; \&=\; a(ei\; -\; fh)\; -\; b(di\; -\; fg)\; +\; c(dh\; -\; eg)\; \backslash \backslash \; \&=\; aei\; +\; bfg\; +\; cdh\; -\; ceg\; -\; bdi\; -\; afh.\; \backslash end$ The rule of Sarrus is a mnemonic for this formula: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration: : This scheme for calculating the determinant of a matrix does not carry over into higher dimensions.''n'' × ''n'' matrices

The Leibniz formula for the determinant of an $n\; \backslash times\; n$-matrix $A$ is a more involved, but related expression. It is an expression involving the notion of ''permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s'' and their ''signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

''. A permutation of the set $\backslash $ is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

$\backslash sigma$ that reorders this set of integers. The value in the $i$-th position after the reordering $\backslash sigma$ is denoted by $\backslash sigma\_i$. The set of all such permutations, the so-called symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

, is denoted $S\_n$. The signature of $\backslash sigma$ is defined to be $+1$ whenever the reordering given by σ can be achieved by successively interchanging two entries an even number of times, and $-1$ whenever it can be achieved by an odd number of such interchanges. Given the matrix $A$ and a permutation $\backslash sigma$, the product
:$a\_\; \backslash cdot\; a\_\; \backslash cdot\; \backslash dots\; \backslash cdot\; a\_$
is also written more briefly using Pi notation as
:$\backslash prod\_^n\; a\_$.
Using these notions, the definition of the determinant using the Leibniz formula is then
:$\backslash det(A)\; =\; \backslash sum\_\; \backslash left(\; \backslash sgn(\backslash sigma)\; \backslash prod\_^n\; a\_\backslash right),$
a sum involving all permutations, where each summand is a product of entries of the matrix, multiplied with a sign depending on the permutation.
The following table unwinds these terms in the case $n=3$. In the first column, a permutation is listed according to its values. For example, in the second row, the permutation $\backslash sigma$ satisfies $\backslash sigma\_1\; =\; 1,\; \backslash sigma\_2\; =\; 3,\; \backslash sigma\_3\; =\; 2$. It can be obtained from the standard order (1, 2, 3) by a single exchange (exchanging the second and third entry), so that its signature is $\backslash sgn(\backslash sigma)=-1$.
The sum of the six terms in the third column then reads
:$\backslash sum\_\; \backslash sgn(\backslash sigma)\; \backslash prod\_^n\; a\_\; =\; a\_a\_a\_\; -\; a\_a\_a\_\; +\; a\_a\_a\_\; -\; a\_a\_a\_\; +\; a\_a\_a\_\; -\; a\_a\_a\_.$
This gives back the formula for $3\; \backslash times\; 3$-matrices above. For a general $n\; \backslash times\; n$-matrix, the Leibniz formula involves $n!$ (''n'' factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

) summands, each of which is a product of ''n'' entries of the matrix.
The Leibniz formula can also be expressed using a summation in which not only permutations, but all sequences of $n$ indices in the range $1,\; \backslash dots,\; n$ occur. To do this, one uses the Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for som ...

$\backslash varepsilon\_$ instead of the sign of a permutation
:$\backslash det(A)\; =\; \backslash sum\_^n\; \backslash varepsilon\_\; a\_\; \backslash cdots\; a\_,$
This gives back the formula above since the Levi-Civita symbol is zero if the indices $i\_1,\; \backslash dots,\; i\_n$ do not form a permutation.
Properties of the determinant

Characterization of the determinant

The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an $n\; \backslash times\; n$-matrix ''A'' as being composed of its $n$ columns, so denoted as :$A\; =\; \backslash big\; (\; a\_1,\; \backslash dots,\; a\_n\; \backslash big\; ),$ where thecolumn vector
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

$a\_i$ (for each ''i'') is composed of the entries of the matrix in the ''i''-th column.
# identity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

.
# linear combination
In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...

$a\_j\; =\; r\; \backslash cdot\; v\; +\; w$ of two column vector
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

s ''v'' and ''w'' and a number ''r'', then the determinant of ''A'' is expressible as a similar linear combination:
#: $\backslash begin,\; A,\; \&=\; \backslash big\; ,\; a\_1,\; \backslash dots,\; a\_,\; r\; \backslash cdot\; v\; +\; w,\; a\_,\; \backslash dots,\; a\_n\; ,\; \backslash \backslash \; \&=\; r\; \backslash cdot\; ,\; a\_1,\; \backslash dots,\; v,\; \backslash dots\; a\_n\; ,\; +\; ,\; a\_1,\; \backslash dots,\; w,\; \backslash dots,\; a\_n\; ,\; \backslash end$
# standard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.
Immediate consequences

These rules have several further consequences: * The determinant is ahomogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar (mathematics), scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneit ...

, i.e., $$\backslash det(cA)\; =\; c^n\backslash det(A)$$ (for an $n\; \backslash times\; n$ matrix $A$).
* Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): $$,\; a\_1,\; \backslash dots,\; a\_j,\; \backslash dots\; a\_i,\; \backslash dots,\; a\_n,\; =\; -\; ,\; a\_1,\; \backslash dots,\; a\_i,\; \backslash dots,\; a\_j,\; \backslash dots,\; a\_n,\; .$$ This formula can be applied iteratively when several columns are swapped. For example $$,\; a\_3,\; a\_1,\; a\_2,\; a\_4\; \backslash dots,\; a\_n,\; =\; -\; ,\; a\_1,\; a\_3,\; a\_2,\; a\_4,\; \backslash dots,\; a\_n,\; =\; ,\; a\_1,\; a\_2,\; a\_3,\; a\_4,\; \backslash dots,\; a\_n,\; .$$ Yet more generally, any permutation of the columns multiplies the determinant by the sign
A sign is an object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Entity, something that is tangible and within the grasp of the senses
** Object (abstract), an object which does not exist at ...

of the permutation.
* If some column can be expressed as a linear combination of the ''other'' columns (i.e. the columns of the matrix form a linearly dependent
In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
* Adding a scalar multiple of one column to ''another'' column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
* If $A$ is a triangular matrix
In the mathematics, mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if al ...

, i.e. $a\_=0$, whenever $i>j$ or, alternatively, whenever $imath>,\; then\; its\; determinant\; equals\; the\; product\; of\; the\; diagonal\; entries:$$\backslash det(A)\; =\; a\_\; a\_\; \backslash cdots\; a\_\; =\; \backslash prod\_^n\; a\_.$$Indeed,\; such\; a\; matrix\; can\; be\; reduced,\; by\; appropriately\; adding\; multiples\; of\; the\; columns\; with\; fewer\; nonzero\; entries\; to\; those\; with\; more\; entries,\; to\; a$diagonal matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

(without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation $\backslash sigma$ which gives a non-zero contribution is the identity permutation.
Example

These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix $A$ using that method: :$A\; =\; \backslash begin\; -2\; \&\; -1\; \&\; 2\; \backslash \backslash \; 2\; \&\; 1\; \&\; 4\; \backslash \backslash \; -3\; \&\; 3\; \&\; -1\; \backslash end.$ Combining these equalities gives $,\; A,\; =\; -,\; E,\; =\; -18\; \backslash cdot\; 3\; \backslash cdot\; (-1)\; =\; 54.$Transpose

The determinant of thetranspose
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces a ...

of $A$ equals the determinant of ''A'':
:$\backslash det\backslash left(A^\backslash textsf\backslash right)\; =\; \backslash det(A)$.
This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an matrix as being composed of ''n'' rows, the determinant is an ''n''-linear function.
Multiplicativity and matrix groups

Thus the determinant is a ''multiplicative map'', i.e., for square matrices $A$ and $B$ of equal size, the determinant of a matrix product equals the product of their determinants: :$\backslash det(AB)\; =\; \backslash det\; (A)\; \backslash det\; (B)$ This key fact can be proven by observing that, for a fixed matrix $B$, both sides of the equation are alternating and multilinear as a function depending on the columns of $A$. Moreover, they both take the value $\backslash det\; B$ when $A$ is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim. A matrix $A$ is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of $\backslash det$ and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by :$\backslash det\backslash left(A^\backslash right)\; =\; \backslash frac\; =;\; href="/html/ALL/s/det(A).html"\; ;"title="det(A)">det(A)$. In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size $n$) forms a group known as thegeneral linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

$\backslash operatorname\_n$ (respectively, a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

called the special linear group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

$\backslash operatorname\_n\; \backslash subset\; \backslash operatorname\_n$. More generally, the word "special" indicates the subgroup of another matrix groupIn mathematics, a matrix group is a group (mathematics), group ''G'' consisting of invertible matrix, invertible matrix (mathematics), matrices over a specified field (mathematics), field ''K'', with the operation of matrix multiplication. A linear g ...

of matrices of determinant one. Examples include the special orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(which if ''n'' is 2 or 3 consists of all rotation matrices), and the special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of Unitary matrix, unitary Matrix (mathematics), matrices with determinant 1.
The more general Unitary group, unitary matrices may have complex determinants with ...

.
The Cauchy–Binet formula is a generalization of that product formula for ''rectangular'' matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.
Laplace expansion

Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . S ...

expresses the determinant of a matrix $A$ in terms of determinants of smaller matrices, known as its minors. The minor $M\_$ is defined to be the determinant of the $(n-1)\; \backslash times\; (n-1)$-matrix that results from $A$ by removing the $i$-th row and the $j$-th column. The expression $(-1)^M\_$ is known as a cofactor. For every $i$, one has the equality
:$\backslash det(A)\; =\; \backslash sum\_^n\; (-1)^\; a\_\; M\_,$
which is called the ''Laplace expansion along the th row''. For example, the Laplace expansion along the first row ($i=1$) gives the following formula:
:$\backslash begina\&b\&c\backslash \backslash \; d\&e\&f\backslash \backslash \; g\&h\&i\backslash end\; =\; a\backslash begine\&f\backslash \backslash \; h\&i\backslash end\; -\; b\backslash begind\&f\backslash \backslash \; g\&i\backslash end\; +\; c\backslash begind\&e\backslash \backslash \; g\&h\backslash end$
Unwinding the determinants of these $2\; \backslash times\; 2$-matrices gives back the Leibniz formula mentioned above. Similarly, the ''Laplace expansion along the $j$-th column'' is the equality
:$\backslash det(A)=\; \backslash sum\_^n\; (-1)^\; a\_\; M\_.$
Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrixIn linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix (math), matrix with the terms of a geometric progression in each row, i.e., an matrix
:V=\begin
1 & x_1 & x_1^2 & \dots & x_1^\\
1 & x_2 & x_2^2 & \ ...

$$\backslash begin\; 1\; \&\; 1\; \&\; 1\; \&\; \backslash cdots\; \&\; 1\; \backslash \backslash \; x\_1\; \&\; x\_2\; \&\; x\_3\; \&\; \backslash cdots\; \&\; x\_n\; \backslash \backslash \; x\_1^2\; \&\; x\_2^2\; \&\; x\_3^2\; \&\; \backslash cdots\; \&\; x\_n^2\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; x\_1^\; \&\; x\_2^\; \&\; x\_3^\; \&\; \backslash cdots\; \&\; x\_n^\; \backslash end\; =\; \backslash prod\_\; \backslash left(x\_j\; -\; x\_i\backslash right).$$
This determinant has been applied, for example, in the proof of Baker's theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithm
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

in the theory of transcendental number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Adjugate matrix

Theadjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...

$\backslash operatorname(A)$ is the transpose of the matrix of the cofactors, that is,
: $(\backslash operatorname(A))\_\; =\; (-1)^\; M\_.$
For every matrix, one has
: $(\backslash det\; A)\; I\; =\; A\backslash operatornameA\; =\; (\backslash operatornameA)\backslash ,A.$
Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

:
: $A^\; =\; \backslash frac\; 1\backslash operatornameA.$
Block matrices

The formula for the determinant of a $2\; \backslash times\; 2$-matrix above continues to hold, under appropriate further assumptions, for ablock matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, i.e., a matrix composed of four submatrices $A,\; B,\; C,\; D$ of dimension $n\; \backslash times\; n$, $n\; \backslash times\; m$, $m\; \backslash times\; n$ and $m\; \backslash times\; m$, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complementIn linear algebra and the theory of matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, sym ...

, is
:$\backslash det\backslash beginA\&\; 0\backslash \backslash \; C\&\; D\backslash end\; =\; \backslash det(A)\; \backslash det(D)\; =\; \backslash det\backslash beginA\&\; B\backslash \backslash \; 0\&\; D\backslash end.$
If $A$ is invertible (and similarly if $D$ is invertible), one has
:$\backslash det\backslash beginA\&\; B\backslash \backslash \; C\&\; D\backslash end\; =\; \backslash det(A)\; \backslash det\backslash left(D\; -\; C\; A^\; B\backslash right)\; .$
If $D$ is a $1\; \backslash times\; 1$-matrix, this simplifies to $\backslash det\; (A)\; (D\; -\; CA^B)$.
If the blocks are square matrices of the ''same'' size further formulas hold. For example, if $C$ and $D$ (i.e., $CD=DC$), then there holds
:$\backslash det\backslash beginA\&\; B\backslash \backslash \; C\&\; D\backslash end\; =\; \backslash det(AD\; -\; BC).$
This formula has been generalized to matrices composed of more than $2\; \backslash times\; 2$ blocks, again under appropriate commutativity conditions among the individual blocks.
For $A\; =\; D$ and $B\; =\; C$, the following formula holds (even if $A$ and $B$ do not commute)
:$\backslash det\backslash beginA\&\; B\backslash \backslash \; B\&\; A\backslash end\; =\; \backslash det(A\; -\; B)\; \backslash det(A\; +\; B).$
Sylvester's determinant theorem

Sylvester's determinant theorem states that for ''A'', an matrix, and ''B'', an matrix (so that ''A'' and ''B'' have dimensions allowing them to be multiplied in either order forming a square matrix): :$\backslash det\backslash left(I\_\backslash mathit\; +\; AB\backslash right)\; =\; \backslash det\backslash left(I\_\backslash mathit\; +\; BA\backslash right),$ where ''I''Sum

The determinant of the sum $A+B$ of two square matrices of the same size is not in general expressible in terms of the determinants of ''A'' and of ''B''. However, for positive semidefinite matrices $A$, $B$ and $C$ of equal size, $\backslash det(A\; +\; B\; +\; C)\; +\; \backslash det(C)\; \backslash geq\; \backslash det(A\; +\; C)\; +\; \backslash det(B\; +\; C)$, for $A,B,C\; \backslash geq\; 0$ with the corollary $\backslash det(A\; +\; B)\; \backslash geq\; \backslash det(A)\; +\; \backslash det(B).$Properties of the determinant in relation to other notions

Eigenvalues and characteristic polynomial

The determinant is closely related to two other central concepts in linear algebra, theeigenvalue
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

s and the characteristic polynomial
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

of a matrix. Let $A$ be an $n\; \backslash times\; n$-matrix with complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

entries with eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to it ...

$\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_n$. (Here it is understood that an eigenvalue with algebraic multiplicity
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

occurs times in this list.) Then the determinant of is the product of all eigenvalues,
:$\backslash det(A)\; =\; \backslash prod\_^n\; \backslash lambda\_i=\backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n.$
The product of all non-zero eigenvalues is referred to as pseudo-determinant.
The characteristic polynomial is defined as
:$\backslash chi\_A(t)\; =\; \backslash det(t\; \backslash cdot\; I\; -\; A).$
Here, $t$ is the indeterminate of the polynomial and $I$ is the identity matrix of the same size as $A$. By means of this polynomial, determinants can be used to find the eigenvalue
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

s of the matrix $A$: they are precisely the root
In vascular plant
Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...

s of this polynomial, i.e., those complex numbers $\backslash lambda$ such that
:$\backslash chi\_A(\backslash lambda)\; =\; 0.$
A Hermitian matrix{{short description, Wikipedia list article
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussi ...

is positive definiteIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

if all its eigenvalues are positive. Sylvester's criterionIn mathematics, Sylvester’s criterion is a necessary and sufficient condition, necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite matrix, positive-definite. It is named after James Joseph Sylvester.
Syl ...

asserts that this is equivalent to the determinants of the submatrices
:$A\_k\; :=\; \backslash begin\; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; a\_\; \&\; a\_\; \&\; \backslash cdots\; \&\; a\_\; \backslash end.$
being positive, for all $k$ between $1$ and $n$.
Trace

Thetrace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band)
Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...

tr(''A'') is by definition the sum of the diagonal entries of and also equals the sum of the eigenvalues. Thus, for complex matrices ,
:$\backslash det(\backslash exp(A))\; =\; \backslash exp(\backslash operatorname(A))$
or, for real matrices ,
:$\backslash operatorname(A)\; =\; \backslash log(\backslash det(\backslash exp(A))).$
Here exp() denotes the matrix exponential
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

of , because every eigenvalue of corresponds to the eigenvalue exp() of exp(). In particular, given any logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of , that is, any matrix satisfying
:$\backslash exp(L)\; =\; A$
the determinant of is given by
:$\backslash det(A)\; =\; \backslash exp(\backslash operatorname(L)).$
For example, for , , and , respectively,
:$\backslash begin\; \backslash det(A)\; \&=\; \backslash frac\backslash left(\backslash left(\backslash operatorname(A)\backslash right)^2\; -\; \backslash operatorname\backslash left(A^2\backslash right)\backslash right),\; \backslash \backslash \; \backslash det(A)\; \&=\; \backslash frac\backslash left(\backslash left(\backslash operatorname(A)\backslash right)^3\; -\; 3\backslash operatorname(A)\; ~\; \backslash operatorname\backslash left(A^2\backslash right)\; +\; 2\; \backslash operatorname\backslash left(A^3\backslash right)\backslash right),\; \backslash \backslash \; \backslash det(A)\; \&=\; \backslash frac\backslash left(\backslash left(\backslash operatorname(A)\backslash right)^4\; -\; 6\backslash operatorname\backslash left(A^2\backslash right)\backslash left(\backslash operatorname(A)\backslash right)^2\; +\; 3\backslash left(\backslash operatorname\backslash left(A^2\backslash right)\backslash right)^2\; +\; 8\backslash operatorname\backslash left(A^3\backslash right)~\backslash operatorname(A)\; -\; 6\backslash operatorname\backslash left(A^4\backslash right)\backslash right).\; \backslash end$
cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, or the Faddeev–LeVerrier algorithm. That is, for generic , the signed constant term of the characteristic polynomial
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

, determined recursively from
:$c\_n\; =\; 1;\; ~~~c\_\; =\; -\backslash frac\backslash sum\_^m\; c\_\; \backslash operatorname\backslash left(A^k\backslash right)\; ~~(1\; \backslash le\; m\; \backslash le\; n)~.$
In the general case, this may also be obtained from
:$\backslash det(A)\; =\; \backslash sum\_\backslash prod\_^n\; \backslash frac\; \backslash operatorname\backslash left(A^l\backslash right)^,$
where the sum is taken over the set of all integers satisfying the equation
:$\backslash sum\_^n\; lk\_l\; =\; n.$
The formula can be expressed in terms of the complete exponential Bell polynomial of ''n'' arguments ''s''Upper and lower bounds

For a positive definite matrix , the trace operator gives the following tight lower and upper bounds on the log determinant :$\backslash operatorname\backslash left(I\; -\; A^\backslash right)\; \backslash le\; \backslash log\backslash det(A)\; \backslash le\; \backslash operatorname(A\; -\; I)$ with equality if and only if . This relationship can be derived via the formula for the KL-divergence between two multivariate normal distributions. Also, :$\backslash frac\; \backslash leq\; \backslash det(A)^\backslash frac\; \backslash leq\; \backslash frac\backslash operatorname(A)\; \backslash leq\; \backslash sqrt.$ These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.Derivative

The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from $\backslash mathbf\; R^$ to $\backslash mathbf\; R$. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula: :$\backslash frac\; =\; \backslash operatorname\backslash left(\backslash operatorname(A)\; \backslash frac\backslash right).$ where $\backslash operatorname(A)$ denotes the adjugate of $A$. In particular, if $A$ is invertible, we have :$\backslash frac\; =\; \backslash det(A)\; \backslash operatorname\backslash left(A^\; \backslash frac\backslash right).$ Expressed in terms of the entries of $A$, these are : $\backslash frac=\; \backslash operatorname(A)\_\; =\; \backslash det(A)\backslash left(A^\backslash right)\_.$ Yet another equivalent formulation is :$\backslash det(A\; +\; \backslash epsilon\; X)\; -\; \backslash det(A)\; =\; \backslash operatorname(\backslash operatorname(A)\; X)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right)\; =\; \backslash det(A)\; \backslash operatorname\backslash left(A^\; X\backslash right)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right)$, using big O notation. The special case where $A\; =\; I$, the identity matrix, yields :$\backslash det(I\; +\; \backslash epsilon\; X)\; =\; 1\; +\; \backslash operatorname(X)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right).$ This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group $\backslash operatorname\_n$ is defined by the equation $\backslash det\; A\; =\; 1$. The above formula shows that its Lie algebra is the special linear Lie algebra $\backslash mathfrak\_n$ consisting of those matrices having trace zero. Writing a $3\; \backslash times\; 3$-matrix as $A\; =\; \backslash begina\; \&\; b\; \&\; c\backslash end$ where $a,\; b,c$ are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two: : $\backslash begin\; \backslash nabla\_\backslash mathbf\backslash det(A)\; \&=\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash \backslash \; \backslash nabla\_\backslash mathbf\backslash det(A)\; \&=\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash \backslash \; \backslash nabla\_\backslash mathbf\backslash det(A)\; \&=\; \backslash mathbf\; \backslash times\; \backslash mathbf.\; \backslash end$History

Historically, determinants were used long before matrices: A determinant was originally defined as a property of asystem of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero).
In this sense, determinants were first used in the Chinese mathematics textbook ''The Nine Chapters on the Mathematical Art'' (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Gerolamo Cardano, Cardano in 1545 by a determinant-like entity.
Determinants proper originated from the work of Seki Takakazu in 1683 in Japan and parallelly of Gottfried Leibniz, Leibniz in 1693. stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of plane curves passing through a given set of points.
Vandermonde (1771) first recognized determinants as independent functions.Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971 gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Joseph Louis Lagrange, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.
Carl Friedrich Gauss, Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a algebraic form, quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.
The next contributor of importance is Jacques Philippe Marie Binet, Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of ''m'' columns and ''n'' rows, which for the special case of reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense, summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. With him begins the theory in its generality.
used the functional determinant which Sylvester later called the Jacobian matrix and determinant, Jacobian. In his memoirs in ''Crelle's Journal'' for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called ''alternants''. About the time of Jacobi's last memoirs, James Joseph Sylvester, Sylvester (1839) and Arthur Cayley, Cayley began their work. introduced the modern notation for the determinant using vertical bars.
The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Otto Hesse, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hermann Hankel, Hankel; circulants by Eugène Charles Catalan, Catalan, William Spottiswoode, Spottiswoode, James Whitbread Lee Glaisher, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Thomas Muir (mathematician), Muir) by Elwin Bruno Christoffel, Christoffel and Ferdinand Georg Frobenius, Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessian matrix, Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
Applications

Cramer's rule

Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as $Ax\; =\; b$. This equation has a unique solution $x$ if and only if $\backslash det\; (A)$ is nonzero. In this case, the solution is given byCramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...

:
:$x\_i\; =\; \backslash frac\; \backslash qquad\; i\; =\; 1,\; 2,\; 3,\; \backslash ldots,\; n$
where $A\_i$ is the matrix formed by replacing the $i$-th column of $A$ by the column vector $b$. This follows immediately by column expansion of the determinant, i.e.
:$\backslash det(A\_i)\; =\; \backslash det\backslash begina\_1\; \&\; \backslash ldots\; \&\; b\; \&\; \backslash ldots\; \&\; a\_n\backslash end\; =\; \backslash sum\_^n\; x\_j\backslash det\backslash begina\_1\; \&\; \backslash ldots\; \&\; a\_\; \&\; a\_j\; \&\; a\_\; \&\; \backslash ldots\; \&\; a\_n\backslash end\; =\; x\_i\backslash det(A)$
where the vectors $a\_j$ are the columns of ''A''. The rule is also implied by the identity
:$A\backslash ,\; \backslash operatorname(A)\; =\; \backslash operatorname(A)\backslash ,\; A\; =\; \backslash det(A)\backslash ,\; I\_n.$
Cramer's rule can be implemented in $\backslash operatorname\; O(n^3)$ time, which is comparable to more common methods of solving systems of linear equations, such as LU decomposition, LU, QR decomposition, QR, or singular value decomposition.
Linear independence

Determinants can be used to characterize linear independence, linearly dependent vectors: $\backslash det\; A$ is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix $A$ are linearly dependent. For example, given two linearly independent vectors $v\_1,\; v\_2\; \backslash in\; \backslash mathbf\; R^3$, a third vector $v\_3$ lies in the Plane (geometry), plane Linear span, spanned by the former two vectors exactly if the determinant of the $3\; \backslash times\; 3$-matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions $f\_1(x),\; \backslash dots,\; f\_n(x)$ (supposed to be $n-1$ times differentiable function, differentiable), the Wronskian is defined to be :$W(f\_1,\; \backslash ldots,\; f\_n)(x)\; =\; \backslash begin\; f\_1(x)\; \&\; f\_2(x)\; \&\; \backslash cdots\; \&\; f\_n(x)\; \backslash \backslash \; f\_1\text{'}(x)\; \&\; f\_2\text{'}(x)\; \&\; \backslash cdots\; \&\; f\_n\text{'}(x)\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; f\_1^(x)\; \&\; f\_2^(x)\; \&\; \backslash cdots\; \&\; f\_n^(x)\; \backslash end.$ It is non-zero (for some $x$) in a specified interval if and only if the given functions and all their derivatives up to order $n-1$ are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See Wronskian#The Wronskian and linear independence, the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root of a function, root.Orientation of a basis

The determinant can be thought of as assigning a number to every sequence of ''n'' vectors in Rstandard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation.
More generally, if the determinant of ''A'' is positive, ''A'' represents an orientation-preserving linear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(if ''A'' is an orthogonal or matrix, this is a rotation (mathematics), rotation), while if it is negative, ''A'' switches the orientation of the basis.
Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of theparallelepiped
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

spanned by those vectors. As a consequence, if $f\; :\; \backslash mathbf\; R^n\; \backslash to\; \backslash mathbf\; R^n$ is the linear map given by multiplication with a matrix $A$, and $S\; \backslash subset\; \backslash mathbf\; R^n$ is any Lebesgue measure, measurable subset, then the volume of $f(S)$ is given by $,\; \backslash det(A),$ times the volume of $S$. More generally, if the linear map $f\; :\; \backslash mathbf\; R^n\; \backslash to\; \backslash mathbf\; R^m$ is represented by the $m\; \backslash times\; n$-matrix $A$, then the $n$-dimensional volume of $f(S)$ is given by:
:$\backslash operatorname(f(S))\; =\; \backslash sqrt\; \backslash operatorname(S).$
By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertex (geometry), vertices $a,\; b,\; c,\; d$, $\backslash frac\; 1\; 6\; \backslash cdot\; ,\; \backslash det(a-b,b-c,c-d),$, or any other combination of pairs of vertices that form a spanning tree over the vertices.
For a general differentiable function, much of the above carries over by considering the Jacobian matrix of ''f''. For
:$f:\; \backslash mathbf\; R^n\; \backslash rightarrow\; \backslash mathbf\; R^n,$
the Jacobian matrix is the matrix whose entries are given by the partial derivatives
:$D(f)\; =\; \backslash left(\backslash frac\; \backslash right)\_.$
Its determinant, the Jacobian determinant
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

, appears in the higher-dimensional version of integration by substitution: for suitable functions ''f'' and an open subset ''U'' of RAbstract algebraic aspects

Determinant of an endomorphism

The above identities concerning the determinant of products and inverses of matrices imply that matrix similarity, similar matrices have the same determinant: two matrices ''A'' and ''B'' are similar, if there exists an invertible matrix ''X'' such that . Indeed, repeatedly applying the above identities yields :$\backslash det(A)\; =\; \backslash det(X)^\; \backslash det(B)\backslash det(X)\; =\; \backslash det(B)\; \backslash det(X)^\; \backslash det(X)\; =\; \backslash det(B).$ The determinant is therefore also called a similarity invariance, similarity invariant. The determinant of alinear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

:$T\; :\; V\; \backslash to\; V$
for some finite-dimensional vector space ''V'' is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis (linear algebra), basis in ''V''. By the similarity invariance, this determinant is independent of the choice of the basis for ''V'' and therefore only depends on the endomorphism ''T''.
Square matrices over commutative rings

The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of acommutative ring
In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative.
Definition and first e ...

$R$, such as the integers $\backslash mathbf\; Z$, as opposed to the field (mathematics), field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies $\backslash det(I)\; =\; 1$ still holds, as do all the properties that result from that characterization.
A matrix $A\; \backslash in\; \backslash operatorname\_(R)$ is invertible (in the sense that there is an inverse matrix whose entries are in $R$) if and only if its determinant is an Unit (ring theory), invertible element in $R$. For $R\; =\; \backslash mathbf\; Z$, this means that the determinant is +1 or −1. Such a matrix is called unimodular matrix, unimodular.
The determinant being multiplicative, it defines a group homomorphism
:$\backslash operatorname\_n(R)\; \backslash rightarrow\; R^\backslash times,$
between the general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

(the group of invertible $n\; \backslash times\; n$-matrices with entries in $R$) and the multiplicative group of units in $R$. Since it respects the multiplication in both groups, this map is a group homomorphism.
Given a ring homomorphism $f\; :\; R\; \backslash to\; S$, there is a map $\backslash operatorname\_n(f)\; :\; \backslash operatorname\_n(R)\; \backslash to\; \backslash operatorname\_n(S)$ given by replacing all entries in $R$ by their images under $f$. The determinant respects these maps, i.e., the identity
:$f(\backslash det((a\_)))\; =\; \backslash det\; ((f(a\_)))$
holds. In other words, the displayed commutative diagram commutes.
For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo $m$ of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo $m$ (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors $\backslash operatorname\_n$ and $(-)^\backslash times$. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,
:$\backslash det:\; \backslash operatorname\_n\; \backslash to\; \backslash mathbb\; G\_m.$
Exterior algebra

The determinant of a linear transformation $T\; :\; V\; \backslash to\; V$ of an $n$-dimensional vector space $V$ or, more generally a free module of (finite) rank of a module, rank $n$ over a commutative ring $R$ can be formulated in a coordinate-free manner by considering the $n$-th exterior algebra, exterior power $\backslash bigwedge^n\; V$ of $V$. The map $T$ induces a linear map :$\backslash begin\; \backslash bigwedge^n\; T:\; \backslash bigwedge^n\; V\; \&\backslash rightarrow\; \backslash bigwedge^n\; V\; \backslash \backslash \; v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n\; \&\backslash mapsto\; T\; v\_1\; \backslash wedge\; T\; v\_2\; \backslash wedge\; \backslash dots\; \backslash wedge\; T\; v\_n.\; \backslash end$ As $\backslash bigwedge^n\; V$ is one-dimensional, the map $\backslash bigwedge^n\; T$ is given by multiplying with some scalar, i.e., an element in $R$. Some authors such as use this fact to ''define'' the determinant to be the element in $R$ satisfying the following identity (for all $v\_i\; \backslash in\; V$): :$\backslash left(\backslash bigwedge^n\; T\backslash right)\backslash left(v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n\backslash right)\; =\; \backslash det(T)\; \backslash cdot\; v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n.$ This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the unicity of a multilinear alternating form on $n$-tuples of vectors in $R^n$. For this reason, the highest non-zero exterior power $\backslash bigwedge^n\; V$ (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of $V$ and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms $\backslash bigwedge^k\; V$ with $k\; <\; n$.Generalizations and related notions

Determinants as treated above admit several variants: the Permanent (mathematics), permanent of a matrix is defined as the determinant, except that the factors $\backslash sgn(\backslash sigma)$ occurring in Leibniz's rule are omitted. The immanant of a matrix, immanant generalizes both by introducing a character theory, character of thesymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

$S\_n$ in Leibniz's rule.
Determinants for finite-dimensional algebras

For any associative algebra $A$ that is dimension, finite-dimensional as a vector space over a field $F$, there is a determinant map :$\backslash det\; :\; A\; \backslash to\; F.$ This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra $A\; =\; \backslash operatorname\_(F)$, but also includes several further cases including the determinant of a quaternion, :$\backslash det\; (a\; +\; ib+jc+kd)\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2$, the Field norm, norm $N\_\; :\; L\; \backslash to\; F$ of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra, also arise as special cases of this construction.Infinite matrices

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators. The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula :$\backslash det(I+A)\; =\; \backslash exp(\backslash operatorname(\backslash log(I+A))).$ Another infinite-dimensional notion of determinant is the functional determinant.Operators in von Neumann algebras

For operators in a finite von Neumann algebra#Factors, factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every State (functional analysis)#tracial state, tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.Related notions for non-commutative rings

For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for , so there is no good definition of the determinant in this setting. For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero with a Regular element (ring theory), regular element of ''R'' as value on some pair of arguments implies that ''R'' is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the ''q''-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of $\backslash mathbb\; Z\_2$-graded rings). Manin matrices form the class closest to matrices with commutative elements.Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques. Computational geometry, however, does frequently use calculations related to determinants. While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating $n!$ ($n$factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

) products for an $n\; \backslash times\; n$-matrix. Thus, the number of required operations grows very quickly: it is Big O notation, of order $n!$. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.
Decomposition methods

Some methods compute $\backslash det(A)$ by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for Positive definite matrix, positive definite matrices). These methods are of order $\backslash operatorname\; O(n^3)$, which is a significant improvement over $\backslash operatorname\; O\; (n!)$. For example, LU decomposition expresses $A$ as a product :$A\; =\; PLU.$ of a permutation matrix $P$ (which has exactly a single $1$ in each column, and otherwise zeros), a lower triangular matrix $L$ and an upper triangular matrix $U$. The determinants of the two triangular matrices $L$ and $U$ can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of $P$ is just the sign $\backslash varepsilon$ of the corresponding permutation (which is $+1$ for an even number of permutations and is $-1$ for an odd number of permutations). Once such a LU decomposition is known for $A$, its determinant is readily computed as :$\backslash det(A)\; =\; \backslash varepsilon\; \backslash det(L)\backslash cdot\backslash det(U).$Further methods

The order $\backslash operatorname\; O(n^3)$ reached by decomposition methods has been improved by different methods. If two matrices of order $n$ can be multiplied in time $M(n)$, where $M(n)\; \backslash ge\; n^a$ for some $a>2$, then there is an algorithm computing the determinant in time $O(M(n))$. This means, for example, that an $\backslash operatorname\; O(n^)$ algorithm exists based on the Coppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373. In addition to the complexity of the algorithm, further criteria can be used to compare algorithms. Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity $\backslash operatorname\; O(n^4)$ is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called closed ordered walks, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule. Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) method is of order $\backslash operatorname\; O(n^3)$, but the bit length of intermediate values can become exponentially long. By comparison, the Bareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times $n$., If the determinant of ''A'' and the inverse of ''A'' have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of , where ''u'' and ''v'' are column vectors. Charles Dodgson (i.e. Lewis Carroll of ''Alice's Adventures in Wonderland'' fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.See also

* Cauchy determinant * Cayley–Menger determinant * Dieudonné determinant * Slater determinantNotes

References

* * * * * * * * * * * * * * * * * * * * G. Baley Price (1947) "Some identities in the theory of determinants", American Mathematical Monthly 54:75–90 * * * * * * *Historical references

* * * * * * * * *External links

* * *Determinant Interactive Program and Tutorial

Linear algebra: determinants.

Compute determinants of matrices up to order 6 using Laplace expansion you choose.

Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.

{{authority control Determinants, Matrix theory Linear algebra Homogeneous polynomials Algebra