コーシー・ビネの公式
det ( A B ) = ∑ π ∈ S m ∑ k : [ m ] → [ n ] i < j ⇒ k ( i ) < k ( j ) A ( 1 ⋯ m k ( π ( 1 ) ) ⋯ k ( π ( m ) ) ) b k ( π ( 1 ) ) , 1 ⋯ b k ( π ( m ) ) , m = ∑ π ∈ S m ∑ k : [ m ] → [ n ] i < j ⇒ k ( i ) < k ( j ) sgn  ( π ) A ( 1 ⋯ m k ( 1 ) ⋯ k ( m ) ) b k ( π ( 1 ) ) , 1 ⋯ b k ( π ( m ) ) , m = ∑ k : [ m ] → [ n ] i < j ⇒ k ( i ) < k ( j ) A ( 1 ⋯ m k ( 1 ) ⋯ k ( m ) ) ∑ π ∈ S m sgn  ( π ) b k ( π ( 1 ) ) , 1 ⋯ b k ( π ( m ) ) , m = ∑ k : [ m ] → [ n ] i < j ⇒ k ( i ) < k ( j ) A ( 1 ⋯ m k ( 1 ) ⋯ k ( m ) ) B ( k ( 1 ) ⋯ k ( m ) 1 ⋯ m ) = ∑ 1 ≤ k 1 < ⋯ < k m ≤ n A ( 1 ⋯ m k 1 ⋯ k m ) B ( k 1 ⋯ k m 1 ⋯ m ) {\displaystyle {\begin{aligned}\det(AB)&=\textstyle \sum \limits _{\pi \in {\mathfrak {S}}_{m}}\sum \limits _{k:[m]\to [n] \atop i<j\Rightarrow k(i)<k(j)}A{\begin{pmatrix}1&\cdots &m\\k(\pi (1))&\cdots &k(\pi (m))\end{pmatrix}}b_{k(\pi (1)),1}\cdots b_{k(\pi (m)),m}\\&=\textstyle \sum \limits _{\pi \in {\mathfrak {S}}_{m}}\sum \limits _{k:[m]\to [n] \atop i<j\Rightarrow k(i)<k(j)}\operatorname {sgn}(\pi )~A{\begin{pmatrix}1&\cdots &m\\k(1)&\cdots &k(m)\end{pmatrix}}b_{k(\pi (1)),1}\cdots b_{k(\pi (m)),m}\\&=\textstyle \sum \limits _{k:[m]\to [n] \atop i<j\Rightarrow k(i)<k(j)}A{\begin{pmatrix}1&\cdots &m\\k(1)&\cdots &k(m)\end{pmatrix}}\sum \limits _{\pi \in {\mathfrak {S}}_{m}}\operatorname {sgn}(\pi )~b_{k(\pi (1)),1}\cdots b_{k(\pi (m)),m}\\&=\textstyle \sum \limits _{k:[m]\to [n] \atop i<j\Rightarrow k(i)<k(j)}A{\begin{pmatrix}1&\cdots &m\\k(1)&\cdots &k(m)\end{pmatrix}}B{\begin{pmatrix}k(1)&\cdots &k(m)\\1&\cdots &m\end{pmatrix}}\\&=\textstyle \sum \limits _{1\leq k_{1}<\cdots <k_{m}\leq n}A{\begin{pmatrix}1&\cdots &m\\k_{1}&\cdots &k_{m}\end{pmatrix}}B{\begin{pmatrix}k_{1}&\cdots &k_{m}\\1&\cdots &m\end{pmatrix}}\end{aligned}}}