# Laplace expansion

This article is about the expansion of the determinant of a square matrix as a weighted sum of determinants of sub-matrices. For the expansion of an 1/r-potential using spherical harmonical functions, see Laplace expansion (potential).

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.

The i, j cofactor of B is the scalar Cij defined by

$C_{ij}\ = (-1)^{i+j} M_{ij}\,,$

where Mij is the i, j minor matrix of B, that is, the determinant of the (n–1) × (n–1) matrix that results from deleting the i-th row and the j-th column of B.

Then the Laplace expansion is given by the following

Theorem. Suppose B = (bij) is an n × n matrix and fix any i, j ∈ {1, 2, ..., n}.

Then its determinant |B| is given by:

$\begin{align}|B| & {} = b_{i1} C_{i1} + b_{i2} C_{i2} + \cdots + b_{in} C_{in} \\ & {} = b_{1j} C_{1j} + b_{2j} C_{2j} + \cdots + b_{nj} C_{nj} \\ & {} = \sum_{j'=1}^{n} b_{ij'} C_{ij'} = \sum_{i'=1}^{n} b_{i'j} C_{i'j} . \end{align}$

## Examples

Consider the matrix

$B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}.$

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:

$|B| = 1 \cdot \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} - 2 \cdot \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 3 \cdot \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix}$
${} = 1 \cdot (-3) - 2 \cdot (-6) + 3 \cdot (-3) = 0.$

Laplace expansion along the second column yields the same result:

$|B| = -2 \cdot \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 5 \cdot \begin{vmatrix} 1 & 3 \\ 7 & 9 \end{vmatrix} - 8 \cdot \begin{vmatrix} 1 & 3 \\ 4 & 6 \end{vmatrix}$
${} = -2 \cdot (-6) + 5 \cdot (-12) - 8 \cdot (-6) = 0.$

It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.

## Proof

Suppose $B$ is an n × n matrix and $i,j\in\{1,2,\dots,n\}.$ For clarity we also label the entries of $B$ that compose its $i,j$ minor matrix $M_{ij}$ as

$(a_{st})$ for $1 \le s,t \le n-1.$

Consider the terms in the expansion of $|B|$ that have $b_{ij}$ as a factor. Each has the form

$\sgn \tau\,b_{1,\tau(1)} \cdots b_{i,j} \cdots b_{n,\tau(n)} = \sgn \tau\,b_{ij} a_{1,\sigma(1)} \cdots a_{n-1,\sigma(n-1)}$

for some permutation τ ∈ Sn with $\tau(i)=j$, and a unique and evidently related permutation $\sigma\in S_{n-1}$ which selects the same minor entries as $\tau.$ Similarly each choice of $\sigma$ determines a corresponding $\tau,$ i.e. the correspondence $\sigma\leftrightarrow\tau$ is a bijection between $S_{n-1}$ and $\{\tau\in S_n\colon\tau(i)=j\}.$ The permutation $\tau$ can be derived from $\sigma$ as follows.

Define $\sigma'\in S_n$ by $\sigma'(k) = \sigma(k)$ for $1 \le k \le n-1$ and $\sigma'(n) = n$. Then $\sgn\sigma'=\sgn\sigma$ and

$\tau\,=(n,n-1,\ldots,i)\sigma'(j,j+1,\ldots,n)$

Since the two cycles can be written respectively as $n-i$ and $n-j$ transpositions,

$\sgn\tau\,= (-1)^{2n-(i+j)} \sgn\sigma'\,= (-1)^{i+j} \sgn\sigma.$

And since the map $\sigma\leftrightarrow\tau$ is bijective,

$\sum_{\tau \in S_n\colon\tau(i)=j}$  $\sgn \tau\,b_{1,\tau(1)} \cdots b_{n,\tau(n)}$
$= \sum_{\sigma \in S_{n-1}} (-1)^{i+j}\sgn\sigma\, b_{ij} a_{1,\sigma(1)} \cdots a_{n-1,\sigma(n-1)}$
$=\ b_{ij} (-1)^{i+j} |M_{ij}|,$

from which the result follows.

## Laplace expansion of a determinant by complementary minors

Laplaces cofactor expansion can be generalised as follows.

### Example

Consider the matrix

$A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix}.$

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 $\left\{1,2,3,4\right\}$, namely let $S=\left\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\right\}$ be the aformentioned set.

By defining the complementary cofactors to be

$b_{\{j,k\}}=\begin{vmatrix} a_{1j} & a_{1k} \\ a_{2j} & a_{2k} \end{vmatrix}$,
$c_{\{j,k\}}=\begin{vmatrix} a_{3j} & a_{3k} \\ a_{4j} & a_{4k} \end{vmatrix}$,

and the sign of their permutation to be

$\varepsilon^{\{i,j\},\{p,q\}}=\mbox{sgn}\begin{bmatrix} 1 & 2 & 3 & 4 \\ i & j & p & q \end{bmatrix}$.

The determinant of B can be written out as

$|B| = \sum_{H \in S} \varepsilon^{H,H^\prime}b_{H}c_{H^\prime},$

where $H^{\prime}$ is the complemenatary set to $H$.

In our explicit example this gives us

$|B| = b_{\{1,2\}}c_{\{3,4\}} -b_{\{1,3\}}c_{\{2,4\}} +b_{\{1,4\}}c_{\{2,3\}} +b_{\{2,3\}}c_{\{1,4\}} -b_{\{2,4\}}c_{\{1,3\}} +b_{\{3,4\}}c_{\{1,2\}}$
${} = \begin{vmatrix} 1 & 2 \\ 5 & 6 \end{vmatrix} \cdot \begin{vmatrix} 11 & 12 \\ 15 & 16 \end{vmatrix} - \begin{vmatrix} 1 & 3 \\ 5 & 7 \end{vmatrix} \cdot \begin{vmatrix} 10 & 12 \\ 14 & 16 \end{vmatrix} + \begin{vmatrix} 1 & 4 \\ 5 & 8 \end{vmatrix} \cdot \begin{vmatrix} 10 & 11 \\ 14 & 15 \end{vmatrix} + \begin{vmatrix} 2 & 3 \\ 6 & 7 \end{vmatrix} \cdot \begin{vmatrix} 9 & 12 \\ 13 & 16 \end{vmatrix} - \begin{vmatrix} 2 & 4 \\ 6 & 8 \end{vmatrix} \cdot \begin{vmatrix} 9 & 11 \\ 13 & 15 \end{vmatrix} + \begin{vmatrix} 3 & 4 \\ 7 & 8 \end{vmatrix} \cdot \begin{vmatrix} 9 & 10 \\ 13 & 14 \end{vmatrix}$
${} = -4 \cdot (-4) -(-8) \cdot (-8) +(-12) \cdot (-4) +(-4) \cdot (-12) -(-8) \cdot (-8) +(-4) \cdot (-4)$
${} = 16 - 64 + 48 + 48 - 64 + 16 = 0.$

As above, It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.

## No good for high dimension

For NxN matrices, the computational effort goes with N!. Therefore, the Laplace expansion is not suitable for large N. Using a decomposition into trigonal matrices, one can determine determinants with effort N3/3.[1]

## References

1. Stoer Bulirsch: Introduction to Numerical Mathematics
• David Poole: Linear Algebra. A Modern Introduction. Cengage Learning 2005, ISBN 0-534-99845-3, p. 265-267 (restricted online copy at Google Books)
• Harvey E. Rose: Linear Algebra. A Pure Mathematical Approach. Springer 2002, ISBN 3-7643-6905-1, p. 57-60 (restricted online copy at Google Books)
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