低次の逆行列

概要

2,3次の逆行列を導きました。

2次

\begin{eqnarray} \left( \begin{array}{cc} a_{00} & a_{01} \\ a_{10} & a_{11} \end{array} \right)^{-1} &=& \frac{1}{D} \left( \begin{array}{cc} a_{11} & -a_{01} \\ -a_{10} & a_{00} \end{array} \right), \\[2mm] && D = a_{00} a_{11}-a_{01} a_{10} \end{eqnarray}

3次

\begin{eqnarray} \left( \begin{array}{cc} a_{00} & a_{01} & a_{02}\\ a_{10} & a_{11} & a_{12}\\ a_{20} & a_{21} & a_{22} \end{array} \right)^{-1} &=& \frac{1}{D} \left( \begin{array}{ccc} a_{11} a_{22}-a_{12} a_{21} & a_{02} a_{21}-a_{01} a_{22} & a_{01} a_{12}-a_{02} a_{11} \\ a_{12} a_{20}-a_{10} a_{22} & a_{00} a_{22}-a_{02} a_{20} & a_{02} a_{10}-a_{00} a_{12} \\ a_{10} a_{21}-a_{11} a_{20} & a_{01} a_{20}-a_{00} a_{21} & a_{00} a_{11}-a_{01} a_{10} \\ \end{array} \right),\\[2mm] && D = a_{00} (a_{11} a_{22}-a_{12} a_{21}) + a_{01} (a_{12}a_{20}-a_{10} a_{22}) + a_{02} (a_{10} a_{21}-a_{11} a_{20}) \end{eqnarray}