498 CHAPTER 27. DETERMINANTS

Example 27.1.4 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 27.1.5 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 appropriate when re-ferring to them. The i jth cofactor of a matrix A will be denoted by cof(A)i j . It is alsoconvenient to refer to the cofactor of an entry of a matrix as follows. For ai j an entry ofthe matrix, its cofactor is just cof(A)i j . Thus the cofactor of the i jth entry is just the i jth

cofactor.

Example 27.1.6 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

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.