19.4. APPLICATIONS 417

the inverse of a matrix in Definition 18.2.25 on Page 390. Also recall the definition of thecofactor matrix given in Definition 19.2.9 on Page 411. This cofactor matrix was just thematrix which results from replacing the i jth entry of the matrix with the i jth cofactor.

The following theorem says that to find the inverse, take the transpose of the cofactormatrix and divide by the determinant. The transpose of the cofactor matrix is called theadjugate or sometimes the classical adjoint of the matrix A. In other words, A−1 is equalto one divided by the determinant of A times the adjugate matrix of A. This is what thefollowing theorem says with more precision. The proof is presented later in the appendixdevoted to the theory of the determinant.

Theorem 19.4.1 A−1 exists if and only if det(A) ̸= 0. If det(A) ̸= 0, then A−1 =(a−1

i j

)where

a−1i j = det(A)−1 cof(A) ji

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

Example 19.4.2 Find the inverse of the matrix

A =

 1 2 33 0 11 2 1

First find the determinant of this matrix. Using Theorems 19.2.20 - 19.2.22 on Page

414, the determinant of this matrix is 12. The cofactor matrix of A is −2 −2 64 −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 33 0 11 2 1

=

 1 0 00 1 00 0 1

and so we got it right. If the result of multiplying these matrices had been something otherthan the identity matrix, you would know there was an error. When this happens, youneed to search for the mistake if you are interested in getting the right answer. A commonmistake is to forget to take the transpose of the cofactor matrix.

This formula for the inverse is also what justifies Cramer’s rule.

Procedure 19.4.3 Suppose A is an n×n matrix and it is desired to solve the systemAx= y,y = (y1, · · · ,yn)

T for x= (x1, · · · ,xn)T . Then Cramer’s rule says

xi =detAi

detA