19.2. AN INTRODUCTION TO DETERMINANTS 411

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 19.2.7 The determinant of a 3× 3 matrix A, is obtained by picking arow (column) and taking the product of each entry in that row (column) with its cofactorand adding these. This process when applied to the ith row (column) is known as expandingthe determinant along the ith row (column).

Example 19.2.8 Find the determinant of

A =

 1 2 34 3 23 2 1

 .

Here is how it is done by “expanding along the first column”.

1

cof(A)11︷ ︸︸ ︷(−1)1+1

∣∣∣∣ 3 22 1

∣∣∣∣+4

cof(A)21︷ ︸︸ ︷(−1)2+1

∣∣∣∣ 2 32 1

∣∣∣∣+3

cof(A)31︷ ︸︸ ︷(−1)3+1

∣∣∣∣ 2 33 2

∣∣∣∣= 0.

This simply follows the rule in the above definition. We took the 1 in the first column andmultiplied it by its cofactor, the 4 in the first column and multiplied it by its cofactor, andthe 3 in the first column and multiplied it by its cofactor. Then we added these numberstogether.

You could also expand the determinant along the second row as follows.

4

cof(A)21︷ ︸︸ ︷(−1)2+1

∣∣∣∣ 2 32 1

∣∣∣∣+3

cof(A)22︷ ︸︸ ︷(−1)2+2

∣∣∣∣ 1 33 1

∣∣∣∣+2

cof(A)23︷ ︸︸ ︷(−1)2+3

∣∣∣∣ 1 23 2

∣∣∣∣= 0.

Observe this gives the same number. You should try expanding along other rows andcolumns. If you don’t make any mistakes, you will always get the same answer.

What about a 4× 4 matrix? You know now how to find the determinant of a 3× 3matrix. The pattern is the same. In general, it is as described in the following definition.

Definition 19.2.9 Let A = (ai j) be an n× n matrix and suppose the determinantof a (n−1)× (n−1) matrix has been defined. Then a new matrix called the cofactormatrix, cof(A) is defined by cof(A)i j = (ci j) where to obtain ci j delete the ith row andthe jth column of A, take the determinant of the (n−1)× (n−1) matrix which results,(This is called the i jth minor of A. ) and then multiply this number by (−1)i+ j. Thus(−1)i+ j ×

(the i jth minor

)equals the i jth cofactor. Then det(A) is given by ∑i Ai jci j =

∑ j Ai jci j. Any of these expansions along a row or a column gives the same number.