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

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:

$\left(\begin{matrix}7 & - 1 \\ 1 & 4\end{matrix}\right)$

Then we take the determinant of this matrix, which is $\left(7 \cdot 4\right) - \left(- 1 \cdot 1\right) = 28 + 1 = 29$. 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, $\left(1 , 1\right)$, and we add them together to get 2. Then we raise $- 1$ to this value, so we get ${\left(- 1\right)}^{2} = 1$, and we multiply this by the minor. So that gives us ${A}_{1 , 1} = 29 \cdot 1 = 29$ 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:

$\left(\begin{matrix}1 & - 2 \\ - 3 & 1\end{matrix}\right)$.

Then we take the determinant of this matrix, which is $\left(1 \cdot 1\right) - \left(- 3 \cdot - 2\right) = 1 - 6 = - 5$. So ${M}_{2 , 3} = - 5$.

To find the cofactor ${A}_{2 , 3}$, we add the subscripts $\left(2 , 3\right)$ to get 5, and raise $- 1$ to this value, which gets us ${\left(- 1\right)}^{5} = - 1$. So we multiply the minor $- 5$ by $- 1$ to get ${A}_{2 , 3} = 5$.