コーシー・ビネの公式
行列式の多重線型性により
det ( A B ) = | ∑ p 1 = 1 n a 1 , p 1 b p 1 , 1 ⋯ ∑ p m = 1 n a 1 , p m b p m , m ⋮ ⋱ ⋮ ∑ p 1 = 1 n a m , p 1 b p 1 , 1 ⋯ ∑ p m = 1 n a m , p m b p m , m | = ∑ p 1 = 1 n ⋯ ∑ p m = 1 n | a 1 , p 1 ⋯ a 1 , p m ⋮ ⋱ ⋮ a m , p 1 ⋯ a m , p m | b p 1 , 1 ⋯ b p m , m = ∑ p : [ m ] → [ n ] A ( 1 ⋯ m p ( 1 ) ⋯ p ( m ) ) b p ( 1 ) , 1 ⋯ b p ( m ) , m {\displaystyle {\begin{aligned}\det(AB)&={\begin{vmatrix}\sum \limits _{p_{1}=1}^{n}a_{1,p_{1}}b_{p_{1},1}&\cdots &\sum \limits _{p_{m}=1}^{n}a_{1,p_{m}}b_{p_{m},m}\\\vdots &\ddots &\vdots \\\sum \limits _{p_{1}=1}^{n}a_{m,p_{1}}b_{p_{1},1}&\cdots &\sum \limits _{p_{m}=1}^{n}a_{m,p_{m}}b_{p_{m},m}\end{vmatrix}}\\&=\textstyle \sum \limits _{p_{1}=1}^{n}\cdots \sum \limits _{p_{m}=1}^{n}{\begin{vmatrix}a_{1,p_{1}}&\cdots &a_{1,p_{m}}\\\vdots &\ddots &\vdots \\a_{m,p_{1}}&\cdots &a_{m,p_{m}}\end{vmatrix}}\,b_{p_{1},1}\cdots b_{p_{m},m}\\&=\textstyle \sum \limits _{p:[m]\to [n]}A{\begin{pmatrix}1&\cdots &m\\p(1)&\cdots &p(m)\end{pmatrix}}b_{p(1),1}\cdots b_{p(m),m}\end{aligned}}}