3次行列式の3行での展開式の導出

${\displaystyle \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=a_{13}{\tilde{a}}_{13}+a_{23}{\tilde{a}}_{23}+a_{33}{\tilde{a}}_{33}}$ ･･････(1)

${\displaystyle ={\left(-1\right)}^{3-1}\left\{\left|\begin{array}{ccc}{a}_{13}& a_{11}& a_{12}\\ 0& a_{21}& a_{22}\\ 0& a_{31}& a_{32}\end{array}\right|+\left|\begin{array}{ccc}0& a_{11}& a_{12}\\ a_{23}& a_{21}& a_{22}\\ a_{33}& a_{31}& a_{32}\end{array}\right|\right\}}$ （行列式の和の性質を用いた）

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

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

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

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

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

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

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

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

${\displaystyle =a_{13}{\tilde{a}}_{13}+a_{23}{\tilde{a}}_{23}+a_{33}{\tilde{a}}_{33}}$

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