Skill: Construct the cofactor matrix

For each entry $a_{ij}$ of a square matrix, delete its row and column to get a smaller matrix. The cofactor $C_{ij}$ is the determinant of that smaller matrix, multiplied by $(-1)^{i+j}$. For a $3\times3$ matrix, the signs follow this checkerboard pattern:

$$\begin{pmatrix} +&-&+\\ -&+&-\\ +&-&+ \end{pmatrix}$$

For example, find the cofactor matrix of

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

Work through the entries in order. For $C_{11}$, delete row 1 and column 1, then take the determinant and use a positive sign:

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

For the rest of the first row:

$$\begin{aligned} C_{12}&=-\begin{vmatrix}3&4\\2&1\end{vmatrix} =-\bigl(3(1)-4(2)\bigr)=5,\\ C_{13}&=+\begin{vmatrix}3&1\\2&-1\end{vmatrix} =3(-1)-1(2)=-5. \end{aligned}$$

Continue in the same way, using the checkerboard signs:

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

Place each cofactor in the same position as its original entry. The cofactor matrix is

$$\begin{pmatrix} 5&5&-5\\ -2&1&5\\ 8&-4&-5 \end{pmatrix}.$$