行列式の和の性質

行列式の1つの行（または列）の各成分が2つの数の和であるならば，その行（または列）を一方の数のみで置き換えた行列式と，他方のみで置き換えた行列式との和になる（多重線形性）．

$3$\displaystyle{ \| \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } &\cdots & \hspace{1}{a}_{ 1n } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ t1 }+\hspace{1}{b}_{ t1 } & \hspace{1}{a}_{ t2 }+\hspace{1}{b}_{ t2 } &\cdots & \hspace{1}{a}_{ tn }+\hspace{1}{b}_{ tn } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 } & \hspace{1}{a}_{ n2 } &\cdots & \hspace{1}{a}_{ nn } & \vspace{6}\\ \end{array} \| }$ $3$\displaystyle{= \| \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } &\cdots & \hspace{1}{a}_{ 1n } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ t1 } & \hspace{1}{a}_{ t2 } &\cdots & \hspace{1}{a}_{ tn } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 } & \hspace{1}{a}_{ n2 } &\cdots & \hspace{1}{a}_{ nn } & \vspace{6}\\ \end{array} \| }$ $3$\displaystyle{+ \| \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } &\cdots & \hspace{1}{a}_{ 1n } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{b}_{ t1 } & \hspace{1}{b}_{ t2 } &\cdots & \hspace{1}{b}_{ tn } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 } & \hspace{1}{a}_{ n2 } &\cdots & \hspace{1}{a}_{ nn } & \vspace{6}\\ \end{array} \| }$

⇒証明へ

■具体例

例：2次の行列式の場合

$3$\displaystyle{ \| \begin{array} \hspace{1}{a}_{1}+\hspace{1}{a}_{2} & \hspace{1}{b}_{1}+\hspace{1}{b}_{2} & \vspace{6}\\ c&d& \vspace{6}\\ \end{array} \| }$

$3$\displaystyle{= $ \hspace{1}{a}_{1}+\hspace{1}{a}_{2} $ d- $ \hspace{1}{b}_{1}+\hspace{1}{b}_{2} $ c}$ $3$\displaystyle{=\hspace{1}{a}_{1}d+\hspace{1}{a}_{2}d- \hspace{1}{b}_{1}c- \hspace{1}{b}_{2}c}$ $3$\displaystyle{=\hspace{1}{a}_{1}d- \hspace{1}{b}_{1}c+\hspace{1}{a}_{2}d- \hspace{1}{b}_{2}c}$

$3$\displaystyle{= \| \begin{array} \hspace{1}{a}_{1} & \hspace{1}{b}_{1} & \vspace{6}\\ c&d& \vspace{6}\\ \end{array} \| + \| \begin{array} \hspace{1}{a}_{2} & \hspace{1}{b}_{2} & \vspace{6}\\ c&d& \vspace{6}\\ \end{array} \| }$

例2：3次の行列式の場合

$3$\displaystyle{ \| \begin{array} \hspace{1}{a}_{1}+\hspace{1}{a}_{2} & \hspace{1}{b}_{1}+\hspace{1}{b}_{2} & \hspace{1}{c}_{1}+\hspace{1}{c}_{2} & \vspace{6}\\ d&e&f& \vspace{6}\\ g&h&i& \vspace{6}\\ \end{array} \| }$

$3$\displaystyle{ = $ \hspace{1}{a}_{1}+ \hspace{1}{a}_{2} $ ef+ $ \hspace{1}{b}_{1}+ \hspace{1}{b}_{2} $ fg }$ $3$\displaystyle{ + $ \hspace{1}{c}_{1}+ \hspace{1}{c}_{2} $ dh- $ \hspace{1}{a}_{1}+ \hspace{1}{a}_{2} $ fh }$ $3$\displaystyle{ + $ \hspace{1}{b}_{1}+ \hspace{1}{b}_{2} $ di+ $ \hspace{1}{c}_{1}+ \hspace{1}{c}_{2} $ eg }$

$3$\displaystyle{ =\hspace{1}{a}_{1}ef+ \hspace{1}{a}_{2}ef+ \hspace{1}{b}_{1}fg+ \hspace{1}{b}_{2}fg }$ $3$\displaystyle{ + \hspace{1}{c}_{1}dh+ \hspace{1}{c}_{2}dh- \hspace{1}{a}_{1}fh- \hspace{1}{a}_{2}fh }$ $3$\displaystyle{ - \hspace{1}{b}_{1}di- \hspace{1}{b}_{2}di- \hspace{1}{c}_{1}eg- \hspace{1}{c}_{2}eg }$

$3$\displaystyle{ =\hspace{1}{a}_{1}ef+ \hspace{1}{b}_{1}fg+ \hspace{1}{c}_{1}dh- \hspace{1}{a}_{1}fh }$ $3$\displaystyle{ - \hspace{1}{b}_{1}di- \hspace{1}{c}_{1}eg+ \hspace{1}{a}_{2}ef+ \hspace{1}{b}_{2}fg }$ $3$\displaystyle{ + \hspace{1}{c}_{2}dh- \hspace{1}{a}_{2}fh- \hspace{1}{b}_{2}di- \hspace{1}{c}_{2}eg }$

$3$\displaystyle{= \| \begin{array} \hspace{1}{a}_{1} & \hspace{1}{b}_{1} & \hspace{1}{c}_{1} & \vspace{6}\\ d&e&f& \vspace{6}\\ g&h&i& \vspace{6}\\ \end{array} \| + \| \begin{array} \hspace{1}{a}_{2} & \hspace{1}{b}_{2} & \hspace{1}{c}_{2} & \vspace{6}\\ d&e&f& \vspace{6}\\ g&h&i& \vspace{6}\\ \end{array} \| }$

例3：2次の行列式の変形例

$3$\displaystyle{ \| \begin{array}1&2& \vspace{6}\\ 3&4& \vspace{6}\\ \end{array} \| = \| \begin{array} 1+ 0 & 0+ 2 & \vspace{6}\\ 3&4& \vspace{6}\\ \end{array} \| }$ $3$\displaystyle{ = \| \begin{array}1&0& \vspace{6}\\ 3&4& \vspace{6}\\ \end{array} \| + \| \begin{array}0&2& \vspace{6}\\ 3&4& \vspace{6}\\ \end{array} \| }$

例4：3次の行列式の変形例

$3$\displaystyle{ \| \begin{array}1&2&3& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array} \| }$

$3$\displaystyle{ = \mid \begin{array} 1+ 0 & 0+ 2 & 0+ 3 & \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}\mid }$ $3$\displaystyle{ = \mid \begin{array}1&0&0& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}\mid + \mid \begin{array}0&2&3& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}\mid }$ $3$\displaystyle{ = \mid \begin{array}1&0&0& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}\mid + \mid \begin{array} 0+ 0 & 2+ 0 & 0+ 3 & \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}\mid }$

$3$\displaystyle{= \| \begin{array}1&0&0& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array} \| + \| \begin{array}0&2&0& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array} \| + \| \begin{array}0&0&3& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array} \| }$

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最終更新日： 2025年4月22日