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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 their changes (cal ...
, the determinant is a scalar value that 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 ...
of the entries 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 (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 ...

linear map
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 :\begin, A, = \begin a & b\\c & d \end=ad-bc.\end Similarly, for a 3 × 3 matrix ''A'', its determinant is :\begin , A, = \begin a & b & c \\ d & e & f \\ g & h & i \end &= a\,\begin e & f \\ h & i \end - b\,\begin d & f \\ g & i \end + c\,\begin d & e \\ g & h \end \\
pt
pt
&= 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 ...

eigenvalue
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 ...

geometry
, the signed -dimensional
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
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 ...

parallelepiped
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 ...

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


First properties

The determinant has several key properties that can be proved by direct evaluation of the definition for 2 \times 2-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of 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 ...

identity matrix
\begin1 & 0 \\ 0 & 1 \end is 1. Second, the determinant is zero if two rows are the same: :\begin a & b \\ a & b \end = ab - ba = 0. This holds similarly if the two columns are the same. Moreover, :\begina & b + b' \\ c & d + d' \end = a(d+d')-(b+b')c = \begina & b\\ c & d \end + \begina & b' \\ c & d' \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: :\begin r \cdot a & b \\ r \cdot c & d \end = rad - brc = r(ad-bc) = r \cdot \begin a & b \\c & d \end.


Geometric meaning

If the matrix entries are real numbers, the matrix can be used to represent two
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 ...

linear map
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 ...

parallelogram
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 ...

identity matrix
). 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). ...

sine
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 ...

cosine
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 : : \text = , \boldsymbol, \,, \boldsymbol, \,\sin\,\theta = \left, \boldsymbol^\perp\\,\left, \boldsymbol\\,\cos\,\theta' = \begin -b \\ a \end \cdot \begin c \\ d \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 ...

bivector
'' 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 = \left begin \mathbf_1 & \mathbf_2 & \cdots & \mathbf_n\end\right/math>, then : A\begin1 \\ 0\\ \vdots \\0\end = \mathbf_1, \quad A\begin0 \\ 1\\ \vdots \\0\end = \mathbf_2, \quad \ldots, \quad A\begin0 \\0 \\ \vdots \\1\end = \mathbf_n. This means that A maps the unit
''n''-cube
''n''-cube
to the ''n''-dimensional parallelotope defined by the vectors \mathbf_1, \mathbf_2, \ldots, \mathbf_n, the region P = \left\. 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, \det(A) = \pm \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: * Means (band) Means was a Canadian Christian Hardcore, Christian post-hardcore and melodic hardcore band from Regina, Saskatchewan, Regina, Saskatchewan. 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 ...

onto
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 ...

one-to-one
, and so is not invertible.


Definition

In the sequel, ''A'' is 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 (geometry), and calculus, change (mathematical analysis, analysis). It ...
with ''n'' rows and ''n'' columns, so that it can be written as :A = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \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: :\begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \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: :\begin \begina&b&c\\d&e&f\\g&h&i\end &= a(ei - fh) - b(di - fg) + c(dh - eg) \\ &= aei + bfg + cdh - ceg - bdi - afh. \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 \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 ...

permutation
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 \ 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 ...
\sigma that reorders this set of integers. The value in the i-th position after the reordering \sigma is denoted by \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 \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 \sigma, the product :a_ \cdot a_ \cdot \dots \cdot a_ is also written more briefly using Pi notation as :\prod_^n a_. Using these notions, the definition of the determinant using the Leibniz formula is then :\det(A) = \sum_ \left( \sgn(\sigma) \prod_^n a_\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 \sigma satisfies \sigma_1 = 1, \sigma_2 = 3, \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 \sgn(\sigma)=-1. The sum of the six terms in the third column then reads : \sum_ \sgn(\sigma) \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 \times 3-matrices above. For a general n \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, \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 ...
\varepsilon_ instead of the sign of a permutation :\det(A) = \sum_^n \varepsilon_ a_ \cdots a_, This gives back the formula above since the Levi-Civita symbol is zero if the indices i_1, \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 \times n-matrix ''A'' as being composed of its n columns, so denoted as :A = \big ( a_1, \dots, a_n \big ), where the
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 ...
a_i (for each ''i'') is composed of the entries of the matrix in the ''i''-th column. #
  • \det\left(I\right) = 1, where I is an
    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 ...

    identity matrix
    . #
  • The determinant is '' multilinear'': if the ''j''th column of a matrix A is written as a
    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 \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: #: \begin, A, &= \big , a_1, \dots, a_, r \cdot v + w, a_, \dots, a_n , \\ &= r \cdot , a_1, \dots, v, \dots a_n , + , a_1, \dots, w, \dots, a_n , \end #
  • The determinant is '' alternating'': whenever two columns of a matrix are identical, its determinant is 0: #: , a_1, \dots, v, \dots, v, \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 \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
    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 a
    homogeneous 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., \det(cA) = c^n\det(A) (for an n \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, \dots, a_j, \dots a_i, \dots, a_n, = - , a_1, \dots, a_i, \dots, a_j, \dots, a_n, . This formula can be applied iteratively when several columns are swapped. For example , a_3, a_1, a_2, a_4 \dots, a_n, = - , a_1, a_3, a_2, a_4, \dots, a_n, = , a_1, a_2, a_3, a_4, \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 ...

    triangular matrix
    , i.e. a_=0, whenever i>j or, alternatively, whenever i, then its determinant equals the product of the diagonal entries: \det(A) = a_ a_ \cdots a_ = \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 \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 = \begin -2 & -1 & 2 \\ 2 & 1 & 4 \\ -3 & 3 & -1 \end. Combining these equalities gives , A, = -, E, = -18 \cdot 3 \cdot (-1) = 54.


    Transpose

    The determinant of the
    transpose 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 ...

    transpose
    of A equals the determinant of ''A'': :\det\left(A^\textsf\right) = \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: :\det(AB) = \det (A) \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 \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 \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 :\det\left(A^\right) = \frac = 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 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 ...
    \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 ...
    \operatorname_n \subset \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) \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 :\det(A) = \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: : \begina&b&c\\ d&e&f\\ g&h&i\end = a\begine&f\\ h&i\end - b\begind&f\\ g&i\end + c\begind&e\\ g&h\end Unwinding the determinants of these 2 \times 2-matrices gives back the Leibniz formula mentioned above. Similarly, the ''Laplace expansion along the j-th column'' is the equality :\det(A)= \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 & \ ...
    \begin 1 & 1 & 1 & \cdots & 1 \\ x_1 & x_2 & x_3 & \cdots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \cdots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^ & x_2^ & x_3^ & \cdots & x_n^ \end = \prod_ \left(x_j - x_i\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

    The
    adjugate 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 ...
    \operatorname(A) is the transpose of the matrix of the cofactors, that is, : (\operatorname(A))_ = (-1)^ M_. For every matrix, one has : (\det A) I = A\operatornameA = (\operatornameA)\,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^ = \frac 1\operatornameA.


    Block matrices

    The formula for the determinant of a 2 \times 2-matrix above continues to hold, under appropriate further assumptions, for a
    block 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 \times n, n \times m, m \times n and m \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 :\det\beginA& 0\\ C& D\end = \det(A) \det(D) = \det\beginA& B\\ 0& D\end. If A is invertible (and similarly if D is invertible), one has :\det\beginA& B\\ C& D\end = \det(A) \det\left(D - C A^ B\right) . If D is a 1 \times 1-matrix, this simplifies to \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 :\det\beginA& B\\ C& D\end = \det(AD - BC). This formula has been generalized to matrices composed of more than 2 \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) :\det\beginA& B\\ B& A\end = \det(A - B) \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): :\det\left(I_\mathit + AB\right) = \det\left(I_\mathit + BA\right), where ''I''''m'' and ''I''''n'' are the and identity matrices, respectively. From this general result several consequences follow.


    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, \det(A + B + C) + \det(C) \geq \det(A + C) + \det(B + C), for A,B,C \geq 0 with the corollary \det(A + B) \geq \det(A) + \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, 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 ...

    eigenvalue
    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 \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 ...

    complex
    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 ...

    eigenvalues
    \lambda_1, \lambda_2, \ldots, \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, :\det(A) = \prod_^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n. The product of all non-zero eigenvalues is referred to as pseudo-determinant. The characteristic polynomial is defined as :\chi_A(t) = \det(t \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 ...

    eigenvalue
    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 \lambda such that :\chi_A(\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 := \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \end. being positive, for all k between 1 and n.


    Trace

    The
    trace 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 , :\det(\exp(A)) = \exp(\operatorname(A)) or, for real matrices , :\operatorname(A) = \log(\det(\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 :\exp(L) = A the determinant of is given by :\det(A) = \exp(\operatorname(L)). For example, for , , and , respectively, :\begin \det(A) &= \frac\left(\left(\operatorname(A)\right)^2 - \operatorname\left(A^2\right)\right), \\ \det(A) &= \frac\left(\left(\operatorname(A)\right)^3 - 3\operatorname(A) ~ \operatorname\left(A^2\right) + 2 \operatorname\left(A^3\right)\right), \\ \det(A) &= \frac\left(\left(\operatorname(A)\right)^4 - 6\operatorname\left(A^2\right)\left(\operatorname(A)\right)^2 + 3\left(\operatorname\left(A^2\right)\right)^2 + 8\operatorname\left(A^3\right)~\operatorname(A) - 6\operatorname\left(A^4\right)\right). \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_ = -\frac\sum_^m c_ \operatorname\left(A^k\right) ~~(1 \le m \le n)~. In the general case, this may also be obtained from :\det(A) = \sum_\prod_^n \frac \operatorname\left(A^l\right)^, where the sum is taken over the set of all integers satisfying the equation :\sum_^n lk_l = n. The formula can be expressed in terms of the complete exponential Bell polynomial of ''n'' arguments ''s''''l'' = −(''l'' – 1)! tr(''A''''l'') as :\det(A) = \frac B_n(s_1, s_2, \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 = \sum_K A^I_K B^K_J, \operatorname(A) = \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, :\det(I + A) = \sum_^\infty \frac \left(-\sum_^\infty \frac \operatorname\left(A^j\right)\right)^k\,, where is the identity matrix. More generally, if :\sum_^\infty \frac \left(-\sum_^\infty \frac\operatorname\left(A^j\right)\right)^k\,, is expanded as a formal power series in then all coefficients of for are zero and the remaining polynomial is .


    Upper and lower bounds

    For a positive definite matrix , the trace operator gives the following tight lower and upper bounds on the log determinant :\operatorname\left(I - A^\right) \le \log\det(A) \le \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, :\frac \leq \det(A)^\frac \leq \frac\operatorname(A) \leq \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 \mathbf R^ to \mathbf R. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula: :\frac = \operatorname\left(\operatorname(A) \frac\right). where \operatorname(A) denotes the adjugate of A. In particular, if A is invertible, we have :\frac = \det(A) \operatorname\left(A^ \frac\right). Expressed in terms of the entries of A, these are : \frac= \operatorname(A)_ = \det(A)\left(A^\right)_. Yet another equivalent formulation is :\det(A + \epsilon X) - \det(A) = \operatorname(\operatorname(A) X) \epsilon + O\left(\epsilon^2\right) = \det(A) \operatorname\left(A^ X\right) \epsilon + O\left(\epsilon^2\right), using big O notation. The special case where A = I, the identity matrix, yields :\det(I + \epsilon X) = 1 + \operatorname(X) \epsilon + O\left(\epsilon^2\right). This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group \operatorname_n is defined by the equation \det A = 1. The above formula shows that its Lie algebra is the special linear Lie algebra \mathfrak_n consisting of those matrices having trace zero. Writing a 3 \times 3-matrix as A = \begina & b & c\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: : \begin \nabla_\mathbf\det(A) &= \mathbf \times \mathbf \\ \nabla_\mathbf\det(A) &= \mathbf \times \mathbf \\ \nabla_\mathbf\det(A) &= \mathbf \times \mathbf. \end


    History

    Historically, determinants were used long before matrices: A determinant was originally defined as a property of 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 ...
    . 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 \det (A) is nonzero. In this case, the solution is given by
    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 ...
    : :x_i = \frac \qquad i = 1, 2, 3, \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. :\det(A_i) = \det\begina_1 & \ldots & b & \ldots & a_n\end = \sum_^n x_j\det\begina_1 & \ldots & a_ & a_j & a_ & \ldots & a_n\end = x_i\det(A) where the vectors a_j are the columns of ''A''. The rule is also implied by the identity :A\, \operatorname(A) = \operatorname(A)\, A = \det(A)\, I_n. Cramer's rule can be implemented in \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: \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 \in \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 \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), \dots, f_n(x) (supposed to be n-1 times differentiable function, differentiable), the Wronskian is defined to be :W(f_1, \ldots, f_n)(x) = \begin f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^(x) & f_2^(x) & \cdots & f_n^(x) \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 R''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
    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). ...
    . 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 the
    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 ...

    parallelepiped
    spanned by those vectors. As a consequence, if f : \mathbf R^n \to \mathbf R^n is the linear map given by multiplication with a matrix A, and S \subset \mathbf R^n is any Lebesgue measure, measurable subset, then the volume of f(S) is given by , \det(A), times the volume of S. More generally, if the linear map f : \mathbf R^n \to \mathbf R^m is represented by the m \times n-matrix A, then the n-dimensional volume of f(S) is given by: :\operatorname(f(S)) = \sqrt \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, \frac 1 6 \cdot , \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: \mathbf R^n \rightarrow \mathbf R^n, the Jacobian matrix is the matrix whose entries are given by the partial derivatives :D(f) = \left(\frac \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 R''n'' (the domain of ''f''), the integral over ''f''(''U'') of some other function is given by :\int_ \phi(\mathbf)\, d\mathbf = \int_U \phi(f(\mathbf)) \left, \det(\operatornamef)(\mathbf)\ \,d\mathbf. The Jacobian also occurs in the inverse function theorem.


    Abstract 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 :\det(A) = \det(X)^ \det(B)\det(X) = \det(B) \det(X)^ \det(X) = \det(B). The determinant is therefore also called a similarity invariance, similarity invariant. The determinant of a
    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 ...
    :T : V \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 a
    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 ...
    R, such as the integers \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 \det(I) = 1 still holds, as do all the properties that result from that characterization. A matrix A \in \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 = \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 :\operatorname_n(R) \rightarrow R^\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 \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 \to S, there is a map \operatorname_n(f) : \operatorname_n(R) \to \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(\det((a_))) = \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 \operatorname_n and (-)^\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, :\det: \operatorname_n \to \mathbb G_m.


    Exterior algebra

    The determinant of a linear transformation T : V \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 \bigwedge^n V of V. The map T induces a linear map :\begin \bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\ v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n. \end As \bigwedge^n V is one-dimensional, the map \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 \in V): :\left(\bigwedge^n T\right)\left(v_1 \wedge \dots \wedge v_n\right) = \det(T) \cdot v_1 \wedge \dots \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 \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 \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 \sgn(\sigma) occurring in Leibniz's rule are omitted. The immanant of a matrix, immanant generalizes both by introducing a character theory, character of the
    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 ...
    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 :\det : A \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 = \operatorname_(F), but also includes several further cases including the determinant of a quaternion, :\det (a + ib+jc+kd) = a^2 + b^2 + c^2 + d^2, the Field norm, norm N_ : L \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 :\det(I+A) = \exp(\operatorname(\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 \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 \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 \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 \operatorname O(n^3), which is a significant improvement over \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 \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 : \det(A) = \varepsilon \det(L)\cdot\det(U).


    Further methods

    The order \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) \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 \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 \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 \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 determinant


    Notes


    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