Skill: Construct the matrix of minors

For each entry of a square matrix, its minor is the determinant left after deleting that entry’s row and column. To form the matrix of minors, put each minor in the same position as the entry it came from. Minors are not given the alternating signs used for cofactors, though a minor’s determinant can itself be negative.

For example, let

$$A=\begin{pmatrix} 1&2&0\\ 3&1&4\\ 2&-1&1 \end{pmatrix}.$$

The minor in position $(1,1)$ comes from deleting row 1 and column 1:

$$M_{11}=\begin{vmatrix}1&4\\-1&1\end{vmatrix}=1(1)-4(-1)=5.$$

For position $(1,2)$, delete row 1 and column 2:

$$M_{12}=\begin{vmatrix}3&4\\2&1\end{vmatrix}=3(1)-4(2)=-5.$$

Repeat this for each position:

$$\begin{aligned} M_{13}&=\begin{vmatrix}3&1\\2&-1\end{vmatrix}=-5, & M_{21}&=\begin{vmatrix}2&0\\-1&1\end{vmatrix}=2, & M_{22}&=\begin{vmatrix}1&0\\2&1\end{vmatrix}=1,\\ M_{23}&=\begin{vmatrix}1&2\\2&-1\end{vmatrix}=-5, & M_{31}&=\begin{vmatrix}2&0\\1&4\end{vmatrix}=8, & M_{32}&=\begin{vmatrix}1&0\\3&4\end{vmatrix}=4,\\ M_{33}&=\begin{vmatrix}1&2\\3&1\end{vmatrix}=-5. \end{aligned}$$

Place the results in their corresponding positions:

$$\text{Matrix of minors}= \begin{pmatrix} 5&-5&-5\\ 2&1&-5\\ 8&4&-5 \end{pmatrix}.$$

Keep each result in the position of the entry whose row and column you deleted.