How to find all the minors and cofactors of the matrix A=((1, -2, 3), ( 6, 7, -1 ), (-3, 1, 4))?

1 Answer
Sep 12, 2015

The minors of a matrix are the determinants of the smaller matrices you get when you delete one row and one column of the original matrix. The cofactors of a matrix are the matrices you get when you multiply the minor by the right sign (positive or negative).

Explanation:

There is a minor and a cofactor for every entry in the matrix -- so that's 9 altogether! I'll just go through a few of them.

The minor for the first row and column is called M_(1,1). We calculate it by removing the first row and column of the matrix, so we are left with the matrix:

((7,-1), (1,4))

Then we take the determinant of this matrix, which is (7*4) - (-1*1) = 28 + 1 = 29. So M_(1,1) = 29.

To find the corresponding cofactor, called A_(1,1), we look at the subscripts on the name of the minor -- in our case, (1,1), and we add them together to get 2. Then we raise -1 to this value, so we get (-1)^2 = 1, and we multiply this by the minor. So that gives us A_(1,1)= 29*1 = 29 as the cofactor.

Let's do one more. To find the minor M_(2,3) we delete the second row and third column of the matrix, so we are left with:

((1, -2),(-3,1)).

Then we take the determinant of this matrix, which is (1*1)-(-3*-2) = 1 - 6 = -5. So M_(2,3) = -5.

To find the cofactor A_(2,3), we add the subscripts (2,3) to get 5, and raise -1 to this value, which gets us (-1)^5 = -1. So we multiply the minor -5 by -1 to get A_(2,3) = 5.