行列式特別な形をした行列式偶数次の交代行列、成分の多項式の累乗(平方)、2次の場合と4次の場合

学習環境

手を動かしてまなぶ線形代数 (藤岡敦(著)、裳華房)の第3章(行列式)、10(特別な形をした行列式)、基本問題の問10.4の解答を求めてみる。

A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]

とおく。

Aが交代行列のとき

\begin{array}{l} A^{T} = - A \\ [\begin{matrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{matrix}] = [\begin{matrix} - a_{11} & - a_{12} \\ - a_{21} & - a_{22} \end{matrix}] \\ a_{11} = - a_{11} \\ a_{11} = 0 \\ a_{21} = - a_{12} \\ a_{22} = - a_{22} \\ a_{22} = 0 \end{array}

よって、

A = [\begin{matrix} 0 & a_{12} \\ - a_{12} & 0 \end{matrix}]

行列式は

\begin{array}{l} \det A \\ = \det [\begin{matrix} 0 & a_{12} \\ - a_{12} & 0 \end{matrix}] \\ = a_{12}^{2} \end{array}

\begin{array}{l} A^{T} = - A \\ a_{j i} = - a_{i j} \\ \det A \\ = \det [\begin{matrix} 0 & a_{12} & a_{13} & a_{14} \\ - a_{12} & 0 & a_{23} & a_{24} \\ - a_{13} & - a_{23} & 0 & a_{34} \\ - a_{14} & - a_{24} & - a_{34} & 0 \end{matrix}] \\ = - a_{12} \det [\begin{matrix} - a_{12} & a_{23} & a_{24} \\ - a_{13} & 0 & a_{34} \\ - a_{14} & - a_{34} & 0 \end{matrix}] + a_{13} \det [\begin{matrix} - a_{12} & 0 & a_{24} \\ - a_{13} & - a_{23} & a_{34} \\ - a_{14} & - a_{24} & 0 \end{matrix}] - a_{14} \det [\begin{matrix} - a_{12} & 0 & a_{23} \\ - a_{13} & - a_{23} & 0 \\ - a_{14} & - a_{24} & - a_{34} \end{matrix}] \\ = - a_{12} (- a_{14} a_{23} a_{34} + a_{13} a_{24} a_{34} - a_{12} a_{34}^{2}) \\ + a_{13} (a_{13} a_{24}^{2} - a_{12} a_{24} a_{34} - a_{14} a_{23} a_{24}) \\ - a_{14} (- a_{12} a_{23} a_{34} + a_{13} a_{23} a_{24} - a_{14} a_{23}^{2}) \\ = a_{34}^{2} a_{12}^{2} + (a_{14} a_{23} a_{34} - a_{13} a_{24} a_{34} - a_{13} a_{24} a_{34} + a_{14} a_{23} a_{34}) a_{12} \\ + a_{13}^{2} a_{24}^{2} - a_{13} a_{14} a_{23} a_{24} - a_{13} a_{14} a_{23} a_{24} + a_{14}^{2} a_{23}^{2} \\ = a_{34}^{2} a_{12}^{2} - 2 (a_{13} a_{24} - a_{14} a_{23}) a_{34} a_{12} + {(a_{13} a_{24} - a_{14} a_{23})}^{2} \\ = {(a_{34} a_{12} - (a_{13} a_{24} - a_{14} a_{23}))}^{2} \\ = {(a_{12} a_{34} - a_{13} a_{24} + a_{14} a_{23})}^{2} \end{array}