3次行列式の3行での展開式の導出

$3$3$ 次の行列式 $3$\displaystyle{\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \hspace{1}{a}_{ 13 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \hspace{1}{a}_{ 23 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \hspace{1}{a}_{ 33 } & \vspace{6}\\ \end{array} }\right|}$ を $3$3$ 列で余因子展開すると

となる．

■導出

行列式の性質を用いて(1)を導出する．

$3$\displaystyle{\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \hspace{1}{a}_{ 13 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \hspace{1}{a}_{ 23 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \hspace{1}{a}_{ 33 } & \vspace{6}\\ \end{array} }\right|}$ $3$\displaystyle{=\left({ - 1 }\right)\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$ （ $3$2$ 列と $3$3$ 列を入れ替えた．）

$3$\displaystyle{={\left({ - 1 }\right)}^{2}\left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$ （ $3$1$ 列と $3$2$ 列を入れ替えた．）

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$ （ $3$\displaystyle{{\left({ - 1 }\right)}^{2}}$ の指数の $3$2$ を行列式の行数 $3$3$ を使って書き直した）

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left|{ \begin{array} \hspace{1}{a}_{ 13 }+0 & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0+\hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0+\hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left{{ \left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\left|{ \begin{array}0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\left|{ \begin{array}0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right| }\right}}$

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left{{ \left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\left({ - 1 }\right)\left|{ \begin{array} \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\left({ - 1 }\right)\left|{ \begin{array}0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right| }\right}}$ （ $3$2$ つ目と行列の $3$1$ 行と $3$2$ 行，および， $3$3$ つ目の行列の $3$2$ 行と $3$3$ 行を入れ替えた）

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left{{ \left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\left({ - 1 }\right)\left|{ \begin{array} \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{ \left({ - 1 }\right) }^{2}\left|{ \begin{array} \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right| }\right}}$ （ $3$3$ つ目の行列の $3$1$ 行と $3$2$ 行を入れ替えた）

$3$\displaystyle{={\left({ - 1 }\right)}^{ 3- 1 }\left{{ { \left({ - 1 }\right) }^{ 1- 1 }\left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{ \left({ - 1 }\right) }^{ 2- 1 }\left|{ \begin{array} \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{ \left({ - 1 }\right) }^{ 3- 1 }\left|{ \begin{array} \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right| }\right}}$

$3$\displaystyle{={\left({ - 1 }\right)}^{ \left({ 1- 1 }\right)+\left({ 3- 1 }\right) }\left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$ $3$\displaystyle{+{\left({ - 1 }\right)}^{ \left({ 2- 1 }\right)+\left({ 3- 1 }\right) }\left|{ \begin{array} \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|}$ $3$\displaystyle{+{\left({ - 1 }\right)}^{ \left({ 3- 1 }\right)+\left({ 3- 1 }\right) }\left|{ \begin{array} \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right|}$

$3$\displaystyle{={\left({ - 1 }\right)}^{ 1+3 }\left|{ \begin{array} \hspace{1}{a}_{ 13 } & \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{\left({ - 1 }\right)}^{ 2+3 }\left|{ \begin{array} \hspace{1}{a}_{ 23 } & \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{\left({ - 1 }\right)}^{ 3+3 }\left|{ \begin{array} \hspace{1}{a}_{ 33 } & \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ 0& \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right|}$

（例えば， $3$\displaystyle{{\left({ - 1 }\right)}^{ \left({ 2- 1 }\right)+\left({ 3- 1 }\right) }={\left({ - 1 }\right)}^{ 2+3- 2 }={\left({ - 1 }\right)}^{ 2+3 }{\left({ - 1 }\right)}^{ - 2 }={\left({ - 1 }\right)}^{ 2+3 }}$ のような計算をした）

$3$\displaystyle{={\left({ - 1 }\right)}^{ 1+3 }\hspace{1}{a}_{ 13 }\left|{ \begin{array} \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{\left({ - 1 }\right)}^{ 2+3 }\hspace{1}{a}_{ 23 }\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+{\left({ - 1 }\right)}^{ 3+3 }\hspace{1}{a}_{ 33 }\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right|}$

次数下げの計算を参照のこと

$3$\displaystyle{=\hspace{1}{a}_{ 13 }{\left({ - 1 }\right)}^{ 1+3 }\left|{ \begin{array} \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\hspace{1}{a}_{ 23 }{\left({ - 1 }\right)}^{ 2+3 }\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|+\hspace{1}{a}_{ 33 }{\left({ - 1 }\right)}^{ 3+3 }\left|{ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \end{array} }\right|}$

$3$\displaystyle{=\hspace{1}{a}_{ 13 }\hspace{1}{\tilde{a}}_{ 13 }+\hspace{1}{a}_{ 23 }\hspace{1}{\tilde{a}}_{ 23 }+\hspace{1}{a}_{ 33 }\hspace{1}{\tilde{a}}_{ 33 }}$

（例えば， $3$\displaystyle{{\left({ - 1 }\right)}^{ 1+3 }\left|{ \begin{array} \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } & \vspace{6}\\ \hspace{1}{a}_{ 31 } & \hspace{1}{a}_{ 32 } & \vspace{6}\\ \end{array} }\right|=\hspace{1}{\tilde{a}}_{ 13 }}$ のように置き換えた，余因子を参照）

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最終更新日： 2025年10月10日