余因子
\(\tilde{a}_{i j}=(-1)^{i+j}\left|\begin{array}{cccccc}a_{11} & \cdots & a_{1(j-1)} & a_{1(j+1)} & \cdots & a_{1 n} \\ \vdots & & \vdots & \vdots & & \\ a_{(i-1) 1} & \cdots & & & \cdots & a_{(i-1) 1} \\ a_{(i+1) 1} & \cdots & & & \cdots & a_{(i+1) 1} \\ \vdots & & \vdots & \vdots & & \\ a_{n 1} & \cdots & a_{n(j-1)} & a_{n(j+1)} & \cdots & a_{n n}\end{array}\right|\)
余因子行列（古典随伴行列）
\(\tilde{A}=\operatorname{adj}(A)=\left[\begin{array}{cccc}\tilde{a}_{11} & \tilde{a}_{21} & \cdots & \tilde{a}_{n 1} \\ \tilde{a}_{12} & \tilde{a}_{22} & \cdots & \tilde{a}_{n 2} \\ \vdots & \vdots & \ddots & \vdots \\ \tilde{a}_{1 n} & \tilde{a}_{2 n} & \cdots & \tilde{a}_{n n}\end{array}\right]\)
行列式の余因子展開
\(\displaystyle \operatorname{det}(A)=a_{1 j} \tilde{a}_{1 j}+a_{2 j} \tilde{a}_{2 j}+\cdots+a_{n j} \tilde{a}_{n j}=\sum_{i=1}^n a_{i j} \tilde{a}_{i j}\)
\(\displaystyle \operatorname{det}(A)=a_{i 1} \tilde{a}_{i 1}+a_{i 2} \tilde{a}_{i 2}+\cdots+a_{i n} \tilde{a}_{i n}=\sum_{j=1}^n a_{i j} \tilde{a}_{i j}\)