10.2. AN INTRODUCTION TO DETERMINANTS 175

Definition 10.2.5 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 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 10.2.6 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

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.

Therefore,

cof(A)23 = (−1)2+3 det

(1 23 2

)= (−1)2+3 (−4) = 4.

Similarly,

cof(A)22 = (−1)2+2 det

(1 33 1

)=−8.

Definition 10.2.7 The determinant of a 3×3 matrix A, is obtained by picking a row (col-umn) and taking the product of each entry in that row (column) with its cofactor and addingthese. This process when applied to the ith row (column) is known as expanding the deter-minant along the ith row (column).

Example 10.2.8 Find the determinant of

A =

 1 2 34 3 23 2 1

 .