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 through matrices.
...
, the Laplace expansion, named after
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
, also called cofactor expansion, is an expression of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of an
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
as a weighted sum of
minors, which are the determinants of some
submatrices
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
of . Specifically, for every ,
where
is the entry of the th row and th column of , and
is the determinant of the submatrix obtained by removing the th row and the th column of .
The term
is called the
cofactor of
in .
The Laplace expansion is often useful in proofs, as in, for example, allowing
recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
on the size of matrices. It is also of didactic interest for its simplicity, and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute, when compared to
Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
.
Examples
Consider the matrix
:
The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields:
:
Laplace expansion along the second column yields the same result:
:
It is easy to verify that the result is correct: the matrix is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
because the sum of its first and third column is twice the second column, and hence its determinant is zero.
Proof
Suppose
is an ''n'' × ''n'' matrix and
For clarity we also label the entries of
that compose its
minor matrix
as
for
Consider the terms in the expansion of
that have
as a factor. Each has the form
:
for some
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
with
, and a unique and evidently related permutation
which selects the same minor entries as . Similarly each choice of determines a corresponding i.e. the correspondence
is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between
and
Using
Cauchy's two-line notation, the explicit relation between
and
can be written as
:
where
is a temporary shorthand notation for a
cycle .
This operation decrements all indices larger than j so that every index fits in the set
The permutation can be derived from as follows.
Define
by
for
and
.
Then
is expressed as
:
Now, the operation which apply
first and then apply
is (Notice applying A before B is equivalent
to applying inverse of A to the upper row of B in two-line notation)
:
where
is temporary shorthand notation for
.
the operation which applies
first and then applies
is
:
above two are equal thus,
:
:
where
is the inverse of
which is
.
Thus
:
Since the two
cycles can be written respectively as
and
transpositions,
:
And since the map
is bijective,
:
from which the result follows. Similarly, the result holds if the index of the outer summation was replaced with
.
Laplace expansion of a determinant by complementary minors
Laplace's cofactor expansion can be generalised as follows.
Example
Consider the matrix
:
The determinant of this matrix can be computed by using the Laplace's cofactor expansion along the first two rows as follows. Firstly note that there are 6 sets of two distinct numbers in namely let
be the aforementioned set.
By defining the complementary cofactors to be
:
:
and the sign of their permutation to be
:
The determinant of ''A'' can be written out as
:
where
is the complementary set to
.
In our explicit example this gives us
:
As above, it is easy to verify that the result is correct: the matrix is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
because the sum of its first and third column is twice the second column, and hence its determinant is zero.
General statement
Let