3.1. BASIC TECHNIQUES AND PROPERTIES 85
This theorem implies the following corollary which gives a way to find determinants. AsI pointed out above, the method of Laplace expansion will not be practical for any matrixof large size.
Corollary 3.1.9 Let A be an n×n matrix and let B be the matrix obtained by replacing theith row (column) of A with the sum of the ith row (column) added to a multiple of anotherrow (column). Then det (A) = det (B) . If B is the matrix obtained from A be replacing theith row (column) of A by a times the ith row (column) then a det (A) = det (B) .
Here is an example which shows how to use this corollary to find a determinant.
Example 3.1.10 Find the determinant of the matrix 1 2 1
1 2 2
1 1 3
First take −1 times the first row and add to the second and the third. The resulting
matrix is 1 2 1
0 0 1
0 −1 2
It has the same determinant as the original matrix. Next switch the bottom two rows toget 1 2 1
0 −1 2
0 0 1
It has determinant which is −1 times the determinant of the original matrix. Hence theoriginal matrix has determinant equal to 1.
The theorem about expanding a matrix along any row or column also provides a way togive a formula for the inverse of a matrix. Recall the definition of the inverse of a matrixin Definition 2.1.22 on Page 48. The following theorem gives a formula for the inverse of amatrix. It is proved in the next section.
Theorem 3.1.11 A−1 exists if and only if det(A) ̸= 0. If det(A) ̸= 0, then A−1 =(a−1ij
)where
a−1ij = det(A)−1 cof (A)ji
for cof (A)ij the ijth cofactor of A.
Theorem 3.1.11 says that to find the inverse, take the transpose of the cofactor matrixand divide by the determinant. The transpose of the cofactor matrix is called the adjugateor sometimes the classical adjoint of the matrix A. It is an abomination to call it the adjointalthough you do sometimes see it referred to in this way. In words, A−1 is equal to one overthe determinant of A times the adjugate matrix of A.
Example 3.1.12 Find the inverse of the matrix
A =
1 2 3
3 0 1
1 2 1