Skill: Calculate cofactors of a matrix

To find the cofactor $C_{ij}$ of an entry in row $i$, column $j$:

1. Delete that entry’s row and column to form its minor.
2. Find the determinant of the minor.
3. Multiply by $(-1)^{i+j}$. The signs follow this checkerboard pattern:

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

For a $2 \times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, its determinant is $ad-bc$.

Worked example: Find every cofactor of

$$A=\begin{bmatrix} 1&2&0\\ 3&-1&4\\ 2&5&1 \end{bmatrix}.$$

For example, to find $C_{12}$, delete row 1 and column 2:

$$\begin{bmatrix} 3&4\\ 2&1 \end{bmatrix}.$$

Its determinant is $3(1)-4(2)=-5$. The checkerboard sign at position $(1,2)$ is negative, so

$$C_{12}=(-1)(-5)=5.$$

Repeat this process for each position:

Cofactor	Minor determinant	Sign	Cofactor value
$C_{11}$	$(-1)(1)-4(5)=-21$	$+$	$-21$
$C_{12}$	$3(1)-4(2)=-5$	$-$	$5$
$C_{13}$	$3(5)-(-1)(2)=17$	$+$	$17$
$C_{21}$	$2(1)-0(5)=2$	$-$	$-2$
$C_{22}$	$1(1)-0(2)=1$	$+$	$1$
$C_{23}$	$1(5)-2(2)=1$	$-$	$-1$
$C_{31}$	$2(4)-0(-1)=8$	$+$	$8$
$C_{32}$	$1(4)-0(3)=4$	$-$	$-4$
$C_{33}$	$1(-1)-2(3)=-7$	$+$	$-7$

Putting the cofactors in their original positions gives the cofactor matrix:

$$\begin{bmatrix} -21&5&17\\ -2&1&-1\\ 8&-4&-7 \end{bmatrix}.$$

Remember: calculate the minor’s determinant first, then apply the checkerboard sign.