基本変形と行列式

【2】数学 Mathematics
【2.2】代数学 Algebra
【2.2.3】線形代数学 Linear algebra
【2.2.3.3】行列式 Determinant
【2.2.3.3.3】行列式の計算

※ このページではMathJax を利用しております。環境によっては数式を表示できないかもしれません。

【P-1】別の列に実数倍加算の証明

ある列を実数倍して、別の列に足しても、行列式は等しいことを証明する。

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}}+r\textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}}+r\textcolor{blue}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}}+r\textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)

線形多重性より、

\( =\left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( + \Large r \normalsize \left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)

同じ列がある行列式について、退化条件より、

\( =\left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)

【P-2】ある列をスカラー倍

これも列の線形多重性の簡易版と考えることができるため、特に計算を要することななく、以下のことが言える。

\( \left| \begin{array}{ccccc} a_{11} & \dots & \textcolor{red}{c}\textcolor{blue}{a_{1j}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{c}\textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{c}\textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( =\Large \textcolor{red}{c} \normalsize \left| \begin{array}{ccccc} a_{11} & \dots & \textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)

【P-3】ある列をスカラー倍

退化条件より、

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize \)

線形多重性より

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( +\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize \)

さらに線形多重性より

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right| \)\( +\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( +\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right| \)\( +\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize \)

退化条件より

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( +\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize \)

よって、

\( \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \)\( \Large = - \normalsize \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right| \)

つまり、行を入れ替えると、符号が逆になる。