行列の積の行列式の定理の証明（3次の正方行列）

3次の正方行列

$A = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}),$ $B = (\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix})$

の行列の積を考える．

$b_{1} = (\begin{matrix} b_{11} & b_{12} & b_{13} \end{matrix})$

$b_{2} = (\begin{matrix} b_{21} & b_{22} & b_{23} \end{matrix})$

$b_{3} = (\begin{matrix} b_{31} & b_{32} & b_{33} \end{matrix})$

とおくと

$B = (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix})$

と表わせる．

また

$A B = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}) (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix})$

$= (\begin{matrix} a_{11} b_{1} + a_{12} b_{2} + a_{13} b_{3} \\ a_{21} b_{1} + a_{22} b_{2} + a_{23} b_{3} \\ a_{31} b_{1} + a_{32} b_{2} + a_{33} b_{3} \end{matrix})$

と表わせる．

次に行列 $A B$ の行列式 $| A B |$ を以下のように変形する．

$| A B | = | \begin{matrix} a_{11} b_{1} + a_{12} b_{2} + a_{13} b_{3} \\ a_{21} b_{1} + a_{22} b_{2} + a_{23} b_{3} \\ a_{31} b_{1} + a_{32} b_{2} + a_{33} b_{3} \end{matrix} |$

$= | \begin{matrix} a_{11} b_{1} \\ a_{21} b_{1} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{21} b_{1} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{21} b_{1} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{11} b_{1} \\ a_{22} b_{2} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{22} b_{2} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{22} b_{2} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{11} b_{1} \\ a_{23} b_{3} \\ a_{33} b_{1} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{23} b_{3} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{11} b_{1} \\ a_{23} b_{3} \\ a_{33} b_{3} \end{matrix} |$

$+ | \begin{matrix} a_{12} b_{2} \\ a_{21} b_{1} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{21} b_{1} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{21} b_{1} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{12} b_{2} \\ a_{22} b_{2} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{22} b_{2} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{22} b_{2} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{12} b_{2} \\ a_{23} b_{3} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{23} b_{3} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{12} b_{2} \\ a_{23} b_{3} \\ a_{33} b_{3} \end{matrix} |$

$+ | \begin{matrix} a_{13} b_{3} \\ a_{21} b_{1} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{21} b_{1} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{21} b_{1} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{13} b_{3} \\ a_{22} b_{2} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{22} b_{2} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{22} b_{2} \\ a_{33} b_{3} \end{matrix} |$ $+ | \begin{matrix} a_{13} b_{3} \\ a_{23} b_{3} \\ a_{31} b_{1} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{23} b_{3} \\ a_{32} b_{2} \end{matrix} | + | \begin{matrix} a_{13} b_{3} \\ a_{23} b_{3} \\ a_{33} b_{3} \end{matrix} |$

$= a_{11} a_{22} a_{33} | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} | + a_{11} a_{23} a_{32} | \begin{matrix} b_{1} \\ b_{3} \\ b_{2} \end{matrix} |$ $+ a_{12} a_{21} a_{33} | \begin{matrix} b_{2} \\ b_{1} \\ b_{3} \end{matrix} | + a_{12} a_{23} a_{31} | \begin{matrix} b_{2} \\ b_{3} \\ b_{1} \end{matrix} |$ $+ a_{13} a_{21} a_{32} | \begin{matrix} b_{3} \\ b_{1} \\ b_{2} \end{matrix} | + a_{13} a_{22} a_{31} | \begin{matrix} b_{3} \\ b_{2} \\ b_{1} \end{matrix} |$

$= a_{11} a_{22} a_{33} \cdot sgn (\begin{matrix} \begin{array}{l} 1 \\ 1 \end{array} & \begin{array}{l} 2 \\ 2 \end{array} & \begin{array}{l} 3 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$ $+ a_{11} a_{23} a_{32} \cdot sgn (\begin{matrix} \begin{array}{l} 1 \\ 1 \end{array} & \begin{array}{l} 3 \\ 2 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$

$+ a_{12} a_{21} a_{33} \cdot sgn (\begin{matrix} \begin{array}{l} 2 \\ 1 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} & \begin{array}{l} 3 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$ $+ a_{12} a_{23} a_{31} \cdot sgn (\begin{matrix} \begin{array}{l} 2 \\ 1 \end{array} & \begin{array}{l} 3 \\ 2 \end{array} & \begin{array}{l} 1 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$

$+ a_{13} a_{21} a_{32} \cdot sgn (\begin{matrix} \begin{array}{l} 3 \\ 1 \end{array} & \begin{array}{l} 1 \\ 2 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$ $+ a_{13} a_{22} a_{31} \cdot sgn (\begin{matrix} \begin{array}{l} 3 \\ 1 \end{array} & \begin{array}{l} 2 \\ 2 \end{array} & \begin{array}{l} 1 \\ 3 \end{array} \end{matrix}) | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$

$= {sgn (\begin{matrix} \begin{array}{l} 1 \\ 1 \end{array} & \begin{array}{l} 2 \\ 2 \end{array} & \begin{array}{l} 3 \\ 3 \end{array} \end{matrix}) a_{11} a_{22} a_{33}$ $+ sgn (\begin{matrix} \begin{array}{l} 1 \\ 1 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} & \begin{array}{l} 3 \\ 2 \end{array} \end{matrix}) a_{11} a_{23} a_{32}$

$+ sgn (\begin{matrix} \begin{array}{l} 1 \\ 2 \end{array} & \begin{array}{l} 2 \\ 1 \end{array} & \begin{array}{l} 3 \\ 3 \end{array} \end{matrix}) a_{12} a_{21} a_{33}$ $+ sgn (\begin{matrix} \begin{array}{l} 1 \\ 2 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} & \begin{array}{l} 3 \\ 1 \end{array} \end{matrix}) a_{12} a_{23} a_{31}$

$+ sgn (\begin{matrix} \begin{array}{l} 1 \\ 3 \end{array} & \begin{array}{l} 2 \\ 1 \end{array} & \begin{array}{l} 3 \\ 2 \end{array} \end{matrix}) a_{13} a_{21} a_{32}$ $+ sgn (\begin{matrix} \begin{array}{l} 1 \\ 3 \end{array} & \begin{array}{l} 2 \\ 2 \end{array} & \begin{array}{l} 3 \\ 1 \end{array} \end{matrix}) a_{13} a_{22} a_{13}} | \begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix} |$

$= {\sum_{3} sgn (\begin{matrix} \begin{array}{l} 1 \\ i_{1} \end{array} & \begin{array}{l} 2 \\ i_{2} \end{array} & \begin{array}{l} 3 \\ i_{3} \end{array} \end{matrix}) a_{1 i_{1}} a_{2 i_{2}} a_{3 i_{3}}} | \begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array} |$

$= | A | | B |$

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最終更新日： 2022年6月18日