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Laplace expansion

The Laplace expansion is a formula for calculating determinants that is calculationally more efficient than the Leibniz formula but less efficient than the decomposition into triangular matrices. See if you like the proof below or the one that heavily relies on permutations.

In fact, we have already dealt with the Laplace expansion when we said that $latex \det A$ is a linear function of column $latex A^{(j)}:$

(1) $latex \det A=\sum_{i=1}^na_{ij}l_{ji}.$

Our task is to analyze the coefficients $latex l_{ji}.$ They will be identified as determinants of certain submatrices of $latex A,$ up to the sign.

Special case

For simplicity, let us start with $latex j=1:$

(2) $latex \det A=a_{11}l_{11}+...+a_{n1}l_{1n}.$

Definition. For any $latex a_{ij},$ denote $latex A_{ij}$ the matrix resulting from $latex A$ after deleting row $latex i$ and column $latex j.$ $latex A_{ij}$ is of size $latex (n-1)\times (n-1).$ Its determinant $latex \det A_{ij}=M_{ij}$ is called $latex (i,j)$-minor.

Step 1. We want to show that $latex l_{11}=M_{11}.$ By the cross-out rule $latex l_{11}$ depends only on the elements of $latex A_{11}.$ When studying $latex l_{11},$ we can assume without loss of generality that

$latex a_{11}=1,$ $latex a_{21}=...=a_{n1}=0.$

Then (2) becomes $latex \det A=l_{11}.$ This allows us to assert that $latex l_{11}$ satisfies Axioms 1-3:

1) $latex l_{11}$ is a homogeneous function of any row of $latex A_{11}$ because $latex \det A$ has this property,

2) Adding one of the rows of $latex A_{11}$ to another row does not change $latex l_{11}$ because $latex \det A$ stays the same, and

3) If $latex A_{11}=I_{(n-1)\times (n-1)},$ then $latex A=I_{n\times n}$ and $latex l_{11}=\det I=1.$

Since, again, $latex l_{11}$ is a function of elements of $latex A_{11}$ only, it follows that

(3) $latex l_{11}=\det A_{11}=M_{11}.$

Step 2. To analyze $latex l_{12},$ as above, we can assume that

$latex a_{21}=1,$ $latex a_{11}=a_{31}=...=a_{n1}=0.$

(2) becomes $latex \det A=l_{12}.$ Here $latex A=(u_2,A^{(2)},...,A^{(n)})$ where $latex u_2$ is the second unit column-vector and $latex A^{(2)},...,A^{(n)}$ are columns of $latex A.$ Let $latex \tilde{A}$ be the result of permuting the first and second rows of $latex A.$ This permutation changes the sign of $latex \det A$ and does not change $latex l_{12}$ (the matrix $latex A_{12}$ for $latex A$ is the same as $latex A_{11}$ for $latex \tilde{A}$). Hence,

(4) $latex \det \tilde{A}=-\det A=-l_{12}.$

For $latex \tilde{A},$ $latex l_{11}$ is the same as $latex l_{11}$ for $latex A$ in Step 1, so (3) and (4) imply $latex l_{12}=-M_{12}.$

Step $latex i.$ We can assume that

$latex a_{i1}=1,$ $latex a_{11}=...=a_{i-1,1}=a_{i+1,1}=a_{n1}=0.$

(2) becomes $latex \det A=l_{1i},$ where $latex A=(u_i,A^{(2)},...,A^{(n)})$ and $latex u_{i}$ is the $latex i$-th unit column-vector. As in Step 2, we can reduce this case to Step 1. Permute rows $latex i$ and $latex i-1,$ then permute rows $latex i-1$ and $latex i-2$ and so on. In total we need $latex i-1$ permutations leading to $latex i-1$ changes in sign. Instead of (4) we get

$latex \det \tilde{A}=(-1)^{i-1}\det A=(-1)^{i-1}l_{1i},$

where $latex \tilde{A}$ is the result of $latex i-1$ permutations. The element in the upper left corner of $latex \tilde{A}$ is 1 and therefore $latex l_{1i}$ is the same as $latex l_{11}$ for $latex A$ in Step 1. The conclusion is that

(5) $latex l_{1i}=(-1)^{i-1}M_{1i}.$

General case

To consider (1), we reduce the case of the $latex j$-th column to the case of the first column. For this, we permute columns $latex j$ and $latex j-1,$ then columns $latex j-1$ and $latex j-2$ and so on. In total we need $latex j-1$ permutations. Denoting the new matrix $latex \tilde{A},$ we have $latex \det \tilde{A}=(-1)^{j-1}\det A.$ To $latex \det \tilde{A}$ we can apply what we know about (2). This will lead to multiplying (5) by $latex (-1)^{j-1}.$ The result is

$latex l_{ji}=(-1)^{i-1+j-1}M_{ji}=(-1)^{i+j}M_{ji}.$

Thus we have derived the Laplace expansion:

Theorem. For $latex j=1,...,n$ one has an expansion by column

$latex \det A=\sum_{i=1}^na_{ij}(-1)^{i+j}M_{ji}.$

The meaning of this expansion is that the calculation of the determinant of $latex A$ is reduced to the calculation of the determinants of matrices of lower dimension. Instead of expansions by column, one can use expansions by row, whichever is convenient. The expression $latex (-1)^{i+j}M_{ji}$ is called a cofactor of the element $latex a_{ij}.$

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