410 CHAPTER 19. EIGENVALUES AND EIGENVECTORS

Example 19.2.2 Find det(

2 4−1 6

).

From the definition this is just (2)(6)− (−1)(4) = 16.Having defined what is meant by the determinant of a 2×2 matrix, what about a 3×3

matrix?

Definition 19.2.3 Suppose A is a 3× 3 matrix. The i jth minor, denoted here asminor(A)i j , is the determinant of the 2× 2 matrix which results from deleting the ith rowand the jth column.

Example 19.2.4 Consider the matrix 1 2 34 3 23 2 1

 .

The (1,2) minor is the determinant of the 2× 2 matrix which results when you delete thefirst row and the second column. This minor is therefore

det(

4 23 1

)=−2.

The (2,3) minor is the determinant of the 2× 2 matrix which results when you delete thesecond row and the third column. This minor is therefore

det(

1 23 2

)=−4.

Definition 19.2.5 Suppose A is a 3× 3 matrix. The i jth cofactor is defined to be(−1)i+ j ×

(i jth minor

). In words, you multiply (−1)i+ j times the i jth minor to get the i jth

cofactor. The cofactors of a matrix are so important that special notation is appropriatewhen referring to them. The i jth cofactor of a matrix A will be denoted by cof(A)i j . It isalso convenient to refer to the cofactor of an entry of a matrix as follows. For ai j an entryof the matrix, its cofactor is just cof(A)i j . Thus the cofactor of the i jth entry is just the i jth

cofactor.

Example 19.2.6 Consider the matrix

A =

 1 2 34 3 23 2 1

 .

The (1,2) minor is the determinant of the 2× 2 matrix which results when you delete thefirst row and the second column. This minor is therefore

det(

4 23 1

)=−2.

It follows

cof(A)12 = (−1)1+2 det(

4 23 1

)= (−1)1+2 (−2) = 2