Minors and cofactors

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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 $\mathbf{A}$.

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

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

Example: Minors and cofactors $\mathbf{A}_e = \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right]$ The minors $M_{1,4}(\mathbf{A}_e)$ and $M_{3,1}(\mathbf{A}_e)$ for example are defined as $M_{1,4}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & \Box & \Box & \Box\\ 3 & 0 & 1 & \Box\\ 0 & 1 & 0 & \Box\\ 0 & 0 & 2 & \Box \end{array}\right|= \left|\begin{array}{ccc} 3 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{array}\right|=6-0=6$ $M_{3,1}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & 2 & 0 & 0\\ \Box & 0 & 1 & 1\\ \Box & \Box & \Box & \Box\\ \Box & 0 & 2 & 1 \end{array}\right|= \left|\begin{array}{ccc} 2 & 0 & 0\\ 0 & 1 & 1\\ 0 & 2 & 1 \end{array}\right|=2-4=-2$ The corresponding cofactors in that case are $C_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6$ $C_{3,1}(\mathbf{A}_e)=(-1)^{3+1}M_{3,1}(\mathbf{A}_e)=(-1)^4\cdot-2=-2$