Cofactor Expansion

Introduction

Cofactor Expansion provides a method for taking the determinant of a matrix which is fairly useful when computations are being done manually.

Given the square matrix , the minor of element a_ij is denoted by M_ij. The minor of an element is defined as the sub-matrix left after the i^th row and j^th column are deleted (crossed out) from . The cofactor of element a_ij, is denoted as C_ij and is equal to (-1)^i+jM_ij and is used to find the sign of a sub-matrix when computing the determinant.

Example

Taking the cofactor of element a₂₁.

It can be shown that the sign of a cofactor is indicated by its position in the matrix .
The sign of the cofactor is based upon the term (1)^i+j

As a result, given the 3 x 3 matrix

Notice that the determinant can be obtained by using cofactors from any single row or column.

Therefore, = a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁ is equivalent to all of the following:

These are known as the cofactor expansion of .

Evaluating the determinants of matrices greater than order 3 x 3 by Cofactor Expansion requires more computational effort than using the elimination method for taking the determinant. However, it can be computed by using either of the following relationships:

This method provides a solution quickly if many zeros are present in a single row or column. This method also provides a recursive way to find the determinant of a square matrix in terms of the determinants of its lower-order submatrices.