Minors and Cofactors

Expansion by minors, used informally above, is stated precisely through two linked ideas: the minor and the cofactor of an entry.

The minor of the entry in row $i$, column $j$, written $M_{ij}$, is the determinant left after deleting row $i$ and column $j$ completely. For $A=\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix},$ the minor of the entry $b_2$ (row 2, column 2) is $M_{22} = \begin{vmatrix} a_1 & c_1 \\ a_3 & c_3 \end{vmatrix},$ obtained by striking out row 2 and column 2.

The cofactor of the same entry, written $C_{ij}$ (or $A_{ij}$), attaches the correct sign to the minor: $C_{ij} = (-1)^{i+j}\, M_{ij}.$ Whenever $i+j$ is even the sign is $+$; whenever $i+j$ is odd it is $-$ — exactly the alternating pattern of the previous section, now written as a formula. So $C_{22} = (-1)^{2+2}M_{22} = +M_{22}$, while $C_{21} = (-1)^{2+1}M_{21} = -M_{21}$.

Expansion along a row (or column) using cofactors is then simply: multiply every entry of that line by its own cofactor and add the products, $|A| = a_i\,C_{i1} + b_i\,C_{i2} + c_i\,C_{i3} \qquad (\text{along row } i).$ This is the precise, general form of the earlier rule, and it works identically for any row or any column — which is why a line already containing a zero is the fastest to expand along: multiplying by zero drops that whole term from the sum with no further work. …