# 行列の積の行列式の定理の証明（n次の正方行列）

n次の正方行列

$A=\left(\begin{array}{cccc}\begin{array}{l}{a}_{11}\\ {a}_{21}\\ {a}_{31}\\ ⋮\\ {a}_{n1}\end{array}& \begin{array}{l}{a}_{12}\\ {a}_{22}\\ {a}_{32}\\ ⋮\\ {a}_{n2}\end{array}& \begin{array}{l}\cdots \\ \cdots \\ \cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{l}{a}_{1n}\\ {a}_{2n}\\ {a}_{3n}\\ ⋮\\ {a}_{nn}\end{array}\end{array}\right),$$B=\left(\begin{array}{cccc}\begin{array}{l}{b}_{11}\\ {b}_{21}\\ {b}_{31}\\ ⋮\\ {a}_{n1}\end{array}& \begin{array}{l}{b}_{12}\\ {b}_{22}\\ {b}_{32}\\ ⋮\\ {a}_{n2}\end{array}& \begin{array}{l}\cdots \\ \cdots \\ \cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{l}{b}_{1n}\\ {b}_{2n}\\ {b}_{3n}\\ ⋮\\ {a}_{nn}\end{array}\end{array}\right)$

${b}_{1}=\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1n}\end{array}\right)$

${b}_{2}=\left(\begin{array}{cccc}{b}_{21}& {b}_{22}& \cdots & {b}_{2n}\end{array}\right)$

${b}_{3}=\left(\begin{array}{cccc}{b}_{31}& {b}_{32}& \cdots & {b}_{3n}\end{array}\right)$

$⋮$

${b}_{n}=\left(\begin{array}{cccc}{b}_{n1}& {b}_{n2}& \cdots & {b}_{nn}\end{array}\right)$

とおくと

$B=\left(\begin{array}{c}{b}_{1}\\ \begin{array}{l}{b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\end{array}\right)$

と表わせる．

また

$AB=\left(\begin{array}{cccc}\begin{array}{l}{a}_{11}\\ {a}_{21}\\ {a}_{31}\\ ⋮\\ {a}_{n1}\end{array}& \begin{array}{l}{a}_{12}\\ {a}_{22}\\ {a}_{32}\\ ⋮\\ {a}_{n2}\end{array}& \begin{array}{l}\cdots \\ \cdots \\ \cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{l}{a}_{1n}\\ {a}_{2n}\\ {a}_{3n}\\ ⋮\\ {a}_{nn}\end{array}\end{array}\right)\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ \begin{array}{l}{b}_{3}\\ ⋮\end{array}\\ {b}_{n}\end{array}\right)$

$=\left(\begin{array}{c}\begin{array}{l}{a}_{11}{b}_{1}+{a}_{12}{b}_{2}+{a}_{13}{b}_{3}+\cdots +{a}_{1n}{b}_{n}\\ {a}_{21}{b}_{1}+{a}_{22}{b}_{2}+{a}_{23}{b}_{3}+\cdots +{a}_{2n}{b}_{n}\end{array}\\ {a}_{31}{b}_{1}+{a}_{32}{b}_{2}+{a}_{33}{b}_{3}+\cdots +{a}_{3n}{b}_{n}\\ ⋮\\ {a}_{n1}{b}_{1}+{a}_{n2}{b}_{2}+{a}_{n3}{b}_{3}+\cdots +{a}_{nn}{b}_{n}\end{array}\right)$

と表わせる．

$\left|AB\right|$$=\left|\begin{array}{c}\begin{array}{l}{a}_{11}{b}_{1}+{a}_{12}{b}_{2}+{a}_{13}{b}_{3}+\cdots +{a}_{1n}{b}_{n}\hfill \\ {a}_{21}{b}_{1}+{a}_{22}{b}_{2}+{a}_{23}{b}_{3}+\cdots +{a}_{2n}{b}_{n}\hfill \end{array}\\ {a}_{31}{b}_{1}+{a}_{32}{b}_{2}+{a}_{33}{b}_{3}+\cdots +{a}_{3n}{b}_{n}\\ ⋮\\ {a}_{n1}{b}_{1}+{a}_{n2}{b}_{2}+{a}_{n3}{b}_{3}+\cdots +{a}_{nn}{b}_{n}\end{array}\right|$

$=\left|\begin{array}{c}{a}_{11}{b}_{1}\\ {a}_{21}{b}_{1}\\ {a}_{31}{b}_{1}\\ ⋮\\ {a}_{n1}{b}_{1}\end{array}\right|+\left|\begin{array}{c}{a}_{11}{b}_{1}\\ {a}_{21}{b}_{1}\\ {a}_{31}{b}_{1}\\ ⋮\\ {a}_{n2}{b}_{2}\end{array}\right|+\cdots +\left|\begin{array}{c}{a}_{11}{b}_{1}\\ {a}_{21}{b}_{1}\\ {a}_{31}{b}_{1}\\ ⋮\\ {a}_{nn}{b}_{n}\end{array}\right|+\cdots \cdots \cdots$ $+\left|\begin{array}{c}{a}_{1n}{b}_{n}\\ {a}_{2n}{b}_{n}\\ {a}_{3\left(n-1\right)}{b}_{n-1}\\ ⋮\\ {a}_{n1}{b}_{1}\end{array}\right|+\left|\begin{array}{c}{a}_{11}{b}_{n}\\ {a}_{2n}{b}_{n}\\ {a}_{3\left(n-1\right)}{b}_{n-1}\\ ⋮\\ {a}_{n2}{b}_{2}\end{array}\right|+\cdots$ $+\left|\begin{array}{c}{a}_{11}{b}_{n}\\ {a}_{21}{b}_{n}\\ {a}_{3\left(n-1\right)}{b}_{n-1}\\ ⋮\\ {a}_{nn}{b}_{n}\end{array}\right|+\left|\begin{array}{c}{a}_{1n}{b}_{n}\\ {a}_{2n}{b}_{n}\\ {a}_{3n}{b}_{n}\\ ⋮\\ {a}_{n1}{b}_{1}\end{array}\right|+\left|\begin{array}{c}{a}_{11}{b}_{n}\\ {a}_{21}{b}_{n}\\ {a}_{3n}{b}_{n}\\ ⋮\\ {a}_{n2}{b}_{2}\end{array}\right|+\cdots +\left|\begin{array}{c}{a}_{11}{b}_{n}\\ {a}_{21}{b}_{n}\\ {a}_{3n}{b}_{n}\\ ⋮\\ {a}_{nn}{b}_{n}\end{array}\right|$

$={a}_{11}{a}_{22}{a}_{33}{a}_{nn}\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|$$+{a}_{12}{a}_{21}{a}_{33}{a}_{nn}\left|\begin{array}{c}{b}_{2}\\ {b}_{1}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|+\cdots \cdots \cdots$ $+{a}_{1n}{a}_{2\left(n-1\right)}{a}_{3\left(n-2\right)}{a}_{n1}\left|\begin{array}{c}{b}_{n}\\ {b}_{n-1}\\ {b}_{n-2}\\ ⋮\\ {b}_{1}\end{array}\right|$

$={a}_{11}{a}_{22}{a}_{33}\cdots {a}_{nn}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ 1& 2& 3& \cdots & n\end{array}\right)\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|$$+{a}_{12}{a}_{21}{a}_{33}\cdots {a}_{nn}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ 2& 1& 3& \cdots & n\end{array}\right)\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|\cdots \cdots \cdots +$$+{a}_{1n}{a}_{2\left(n-1\right)}{a}_{3\left(n-2\right)}\cdots {a}_{n1}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ n& n-1& n-2& \cdots & 1\end{array}\right)\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|$

$=\left\{{a}_{11}{a}_{22}{a}_{33}\cdots {a}_{nn}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ 1& 2& 3& \cdots & n\end{array}\right)\right\$$+{a}_{12}{a}_{21}{a}_{33}\cdots {a}_{nn}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ 2& 1& 3& \cdots & n\end{array}\right)\cdots \cdots \cdots$$+{a}_{1n}{a}_{2\left(n-1\right)}{a}_{3\left(n-2\right)}\cdots {a}_{n1}·\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ n& n-1& n-2& \cdots & 1\end{array}\right)}\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|$

$=\left\{\sum _{3}\mathrm{sgn}\left(\begin{array}{ccccc}1& 2& 3& \cdots & n\\ {i}_{1}& {i}_{2}& {i}_{13}& \cdots & {i}_{n}\end{array}\right){a}_{1{i}_{1}}{a}_{2{i}_{2}}{a}_{3{i}_{3}}\cdots {a}_{n{i}_{n}}\right\}\left|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{n}\end{array}\right|$

$=|A||B|$

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