Minors and cofactors

The minor $M_{i,j}(\mathbf{A})$ of an n-by-n square matrix $\mathbf{A}$ is the determinant of a smaller square matrix obtained by removing the row $i$ and the column $j$ from $\mathbf{A}$.

{{Example |Title=Minors

The minors $M_{1,4}(\mathbf{A}_e)$ and $M_{3,1}(\mathbf{A}_e)$ for example are defined as

$M_{1,4}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & \Box & \Box & \Box\\ 3 & 0 & 1 & \Box\\ 0 & 1 & 0 & \Box\\ 0 & 0 & 2 & \Box \end{array}\right|= \left|\begin{array}{ccc} 3 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{array}\right|=6-0=6$

 $M_{3,1}(\mathbf{A}_e)=
\left|\begin{array}{cccc}
\Box & 2 & 0 & 0\\
\Box & 0 & 1 & 1\\
\Box & \Box & \Box & \Box\\
\Box & 0 & 2 & 1
\end{array}\right|=
\left|\begin{array}{ccc}
2 & 0 & 0\\
0 & 1 & 1\\
0 & 2 & 1
\end{array}\right|=2-4=-2$

Multiplying the minor with $(-1)^{i+j}$ results in the cofactor $C_{i,j}(\mathbf{A})$:

$C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{i,j}(\mathbf{A})$

 $C_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6$