Question

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

Answer 1

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`.