182 CHAPTER 10. EIGENVALUES AND EIGENVECTORS

Theorem 10.3.1 A−1 exists if and only if det(A) ̸= 0. If det(A) ̸= 0, then A−1 =(

a−1i j

)where

a−1i j = det(A)−1 cof(A) ji

for cof(A)i j the i jth cofactor of A.

Example 10.3.2 Find the inverse of the matrix

A =

 1 2 33 0 11 2 1

First find the determinant of this matrix. Using Theorems 10.2.20 - 10.2.22 on Page

178, the determinant of this matrix equals the determinant of the matrix 1 2 30 −6 −80 0 −2

which equals 12. The cofactor matrix of A is −2 −2 6

4 −2 02 8 −6

 .

Each entry of A was replaced by its cofactor. Therefore, from the above theorem, theinverse of A should equal

112

 −2 −2 64 −2 02 8 −6

T

=

 −1/6 1/3 1/6−1/6 −1/6 2/31/2 0 −1/2

 .

Does it work? You should check to see if it does. When the matrices are multiplied −1/6 1/3 1/6−1/6 −1/6 2/31/2 0 −1/2

 1 2 3

3 0 11 2 1

=

 1 0 00 1 00 0 1

and so it is correct.

Example 10.3.3 Find the inverse of the matrix

A =



12 0 1

2

− 16

13 − 1

2

− 56

23 − 1

2

