How to Solve Matrix Determinants

Follow this example 3x3 matrix A. A equals:

9 5 -3

2 7 1

0 3 5

Select a single row or column of the matrix. In the example the top row is taken:

9 5 -3

Find the minor matrices of each of the elements of the selected row. Remove the row and column that the particular element lies in and isolate the remaining 2x2 matrix. In the example the remaining 2x2 matrix of the first element in the selected row (9) is:

7 1

3 5

The remaining 2x2 matrix of the second element in the selected row (5) is:

2 1

0 5

The remaining 2x2 matrix of the third element in the selected row (-3) is:

2 7

0 3

Find the determinants of the isolated 2x2 matrices. These determinants are the minors of the corresponding elements. The minor of the first element in the example row (9) is:

7*5 - 1*3 = 32

The minor of the second element in the example row (5) is:

2*5 - 1*0 = 10

The minor of the third element in the example row (-3) is:

2*3 - 7*0 = 6

Multiply each of the minors found in step 3 by (-1)^(i + j) where i is the row of the element and j is the column of the element. This gives you the cofactor of each of the elements in the example row. The cofactor of the first element in the example row (9) is:

((-1)^(1+1))*32 = 32

The cofactor of the second element in the example row (5) is:

((-1)^(1+2))*10 = -10

The cofactor of the third element in the example row (-3) is:

((-1)^(1+3))*6 = 6

Multiply each of the cofactors by their corresponding elements and then add them all together. This solves the determinant:

32*9 +(-10)*5 + 6*(-3) = 220

In the example the determinant of the matrix is 220.