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余因子

${\tilde{a}}_{ij}=(-1{)}^{i+j}\left|\begin{array}{cccccc}{a}_{11}& \cdots & a_{1(j-1)}& a_{1(j+1)}& \cdots & a_{1n}\\ ⋮& & ⋮& ⋮& & \\ a_{(i-1)1}& \cdots & & & \cdots & a_{(i-1)1}\\ a_{(i+1)1}& \cdots & & & \cdots & a_{(i+1)1}\\ ⋮& & ⋮& ⋮& & \\ a_{n1}& \cdots & a_{n(j-1)}& a_{n(j+1)}& \cdots & a_{nn}\end{array}\right|$

余因子行列（古典随伴行列）

$\tilde{A}=\mathrm{adj}(A)=\left[\begin{array}{cccc}{\tilde{a}}_{11}& {\tilde{a}}_{21}& \cdots & {\tilde{a}}_{n1}\\ {\tilde{a}}_{12}& {\tilde{a}}_{22}& \cdots & {\tilde{a}}_{n2}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{a}}_{1n}& {\tilde{a}}_{2n}& \cdots & {\tilde{a}}_{nn}\end{array}\right]$

行列式の余因子展開

${\displaystyle \det (A)=a_{1j}{\tilde{a}}_{1j}+a_{2j}{\tilde{a}}_{2j}+\cdots +a_{nj}{\tilde{a}}_{nj}=\sum _{i=1}^n{a}_{ij}{\tilde{a}}_{ij}}$

${\displaystyle \det (A)=a_{i1}{\tilde{a}}_{i1}+a_{i2}{\tilde{a}}_{i2}+\cdots +a_{in}{\tilde{a}}_{in}=\sum _{j=1}^n{a}_{ij}{\tilde{a}}_{ij}}$