116 CHAPTER 6. DETERMINANTS

which equals 12. The cofactor matrix of A is −2 −2 64 −2 02 8 −6

 .

Each entry of A was replaced by the cofactor associated with the position of the entry.Therefore, from the above theorem, the inverse 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 6.2.3 Find the inverse of the matrix

A =



12 0 1

2

− 16

13 − 1

2

− 56

23 − 1

2

First find its determinant. This determinant is 1

6 . Now replace each entry by the cofactor

associated with the position of the entry. Thus the cofactor associated with the − 16 in the

first column is −det

(0 1/2

2/3 −1/2

). After this, take the transpose of what results and

multiply by 6 which is 1/(det(A)). Thus, the inverse is

6

16

13

16

13

16 − 1

3− 1

616

16

T

.

Then

6

 1/6 1/3 1/61/3 1/6 −1/3−1/6 1/6 1/6

T

=

 1 2 −12 1 11 −2 1

which should be the inverse. Always check your work. 1 2 −1

2 1 11 −2 1

 1/2 0 1/2−1/6 1/3 −1/2−5/6 2/3 −1/2

=

 1 0 00 1 00 0 1

