低次の連立1次方程式の解公式

概要

2～4変数の連立1次方程式の解公式を導きました。

2変数

連立1次方程式 \begin{eqnarray} \left( \begin{array}{cc} a_{00} & a_{01} \\ a_{10} & a_{11} \end{array} \right) \left( \begin{array}{c} x_0 \\ x_1 \end{array} \right) &=& \left( \begin{array}{c} b_0 \\ b_1 \end{array} \right) \end{eqnarray} の解は \begin{eqnarray} D &=& a_{00} a_{11}-a_{01} a_{10} \end{eqnarray} が 0 でないとして \begin{eqnarray} x_0 &=& \frac{1}{D} (a_{11} b_{0}-a_{01} b_{1}) \\ x_1 &=& \frac{1}{D} (a_{00} b_{1}-a_{10} b_{0}) \end{eqnarray}

3変数

連立1次方程式 \begin{eqnarray} \left( \begin{array}{cc} a_{00} & a_{01} & a_{02}\\ a_{10} & a_{11} & a_{12}\\ a_{10} & a_{11} & a_{22} \end{array} \right) \left( \begin{array}{c} x_0 \\ x_1 \\ x_2 \end{array} \right) &=& \left( \begin{array}{c} b_0 \\ b_1 \\ b_2 \end{array} \right) \end{eqnarray} の解は \begin{eqnarray} 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} が 0 でないとして \begin{eqnarray} x_0 &=& \frac{1}{D} \left\{ b_{0}(a_{11} a_{22}-a_{12} a_{21}) + b_{1}(a_{02} a_{21}-a_{01} a_{22}) + b_{2}(a_{01} a_{12}-a_{02} a_{11}) \right\}\\ x_1 &=& \frac{1}{D} \left\{ b_{0}(a_{12} a_{20}-a_{10} a_{22}) + b_{1}(a_{00} a_{22}-a_{02} a_{20}) + b_{2}(a_{02} a_{10}-a_{00} a_{12}) \right\} \\ x_2 &=& \frac{1}{D} \left\{ b_{0}(a_{10} a_{21}-a_{11} a_{20}) + b_{1}(a_{01} a_{20}-a_{00} a_{21}) + b_{2}(a_{00} a_{11}-a_{01} a_{10}) \right\} \end{eqnarray}

4変数

連立1次方程式 \begin{eqnarray} \left( \begin{array}{cc} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \\ \end{array} \right) \left( \begin{array}{c} x_0 \\ x_1 \\ x_2 \\ x_3 \end{array} \right) &=& \left( \begin{array}{c} b_0 \\ b_1 \\ b_2 \\ b_3 \end{array} \right) \end{eqnarray} の解は \begin{eqnarray} D &=&\ \ \ a_{00} \left\{a_{11} (a_{22} a_{33}-a_{23} a_{32})+a_{12} (a_{23} a_{31}-a_{21} a_{33})+a_{13} (a_{21} a_{32}-a_{22} a_{31})\right\} \nonumber\\ &&+a_{01} \left\{a_{10} (a_{23} a_{32}-a_{22} a_{33})+a_{12} (a_{20} a_{33}-a_{23} a_{30})+a_{13} (a_{22} a_{30}-a_{20} a_{32})\right\} \nonumber\\ &&+a_{02} \left\{a_{10} (a_{21} a_{33}-a_{23} a_{31})+a_{11} (a_{23} a_{30}-a_{20} a_{33})+a_{13} (a_{20} a_{31}-a_{21} a_{30})\right\} \nonumber\\ &&+a_{03} \left\{a_{10} (a_{22} a_{31}-a_{21} a_{32})+a_{11} (a_{20} a_{32}-a_{22} a_{30})+a_{12} (a_{21} a_{30}-a_{20} a_{31})\right\} \end{eqnarray} が 0 でないとして \begin{eqnarray} x_0 &=& \frac{1}{D} \left[ \begin{array}{l} \ \ \ b_0 \left\{a_{11} (a_{22} a_{33}-a_{23} a_{32})+a_{12} (a_{23} a_{31}-a_{21} a_{33})+a_{13} (a_{21} a_{32}-a_{22} a_{31})\right\}\\ +b_1 \left\{a_{01} (a_{23} a_{32}-a_{22} a_{33})+a_{02} (a_{21} a_{33}-a_{23} a_{31})+a_{03} (a_{22} a_{31}-a_{21} a_{32})\right\}\\ +b_2 \left\{a_{01} (a_{12} a_{33}-a_{13} a_{32})+a_{02} (a_{13} a_{31}-a_{11} a_{33})+a_{03} (a_{11} a_{32}-a_{12} a_{31})\right\}\\ +b_3 \left\{a_{01} (a_{13} a_{22}-a_{12} a_{23})+a_{02} (a_{11} a_{23}-a_{13} a_{21})+a_{03} (a_{12} a_{21}-a_{11} a_{22})\right\} \end{array} \right] \end{eqnarray} \begin{eqnarray} x_1 &=& \frac{1}{D} \left[ \begin{array}{l} \ \ \ b_0 \left\{a_{10} (a_{23} a_{32}-a_{22} a_{33})+a_{12} (a_{20} a_{33}-a_{23} a_{30})+a_{13} (a_{22} a_{30}-a_{20} a_{32})\right\}\\ +b_1 \left\{a_{00} (a_{22} a_{33}-a_{23} a_{32})+a_{02} (a_{23} a_{30}-a_{20} a_{33})+a_{03} (a_{20} a_{32}-a_{22} a_{30})\right\}\\ +b_2 \left\{a_{00} (a_{13} a_{32}-a_{12} a_{33})+a_{02} (a_{10} a_{33}-a_{13} a_{30})+a_{03} (a_{12} a_{30}-a_{10} a_{32})\right\}\\ +b_3 \left\{a_{00} (a_{12} a_{23}-a_{13} a_{22})+a_{02} (a_{13} a_{20}-a_{10} a_{23})+a_{03} (a_{10} a_{22}-a_{12} a_{20})\right\} \end{array} \right] \end{eqnarray} \begin{eqnarray} x_2 &=& \frac{1}{D} \left[ \begin{array}{l} \ \ \ b_0 \left\{a_{10} (a_{21} a_{33}-a_{23} a_{31})+a_{11} (a_{23} a_{30}-a_{20} a_{33})+a_{13} (a_{20} a_{31}-a_{21} a_{30})\right\}\\ +b_1 \left\{a_{00} (a_{23} a_{31}-a_{21} a_{33})+a_{01} (a_{20} a_{33}-a_{23} a_{30})+a_{03} (a_{21} a_{30}-a_{20} a_{31})\right\}\\ +b_2 \left\{a_{00} (a_{11} a_{33}-a_{13} a_{31})+a_{01} (a_{13} a_{30}-a_{10} a_{33})+a_{03} (a_{10} a_{31}-a_{11} a_{30})\right\}\\ +b_3 \left\{a_{00} (a_{13} a_{21}-a_{11} a_{23})+a_{01} (a_{10} a_{23}-a_{13} a_{20})+a_{03} (a_{11} a_{20}-a_{10} a_{21})\right\} \end{array} \right] \end{eqnarray} \begin{eqnarray} x_3 &=& \frac{1}{D} \left[ \begin{array}{l} \ \ \ b_0 \left\{a_{10} (a_{22} a_{31}-a_{21} a_{32})+a_{11} (a_{20} a_{32}-a_{22} a_{30})+a_{12} (a_{21} a_{30}-a_{20} a_{31})\right\}\\ +b_1 \left\{a_{00} (a_{21} a_{32}-a_{22} a_{31})+a_{01} (a_{22} a_{30}-a_{20} a_{32})+a_{02} (a_{20} a_{31}-a_{21} a_{30})\right\}\\ +b_2 \left\{a_{00} (a_{12} a_{31}-a_{11} a_{32})+a_{01} (a_{10} a_{32}-a_{12} a_{30})+a_{02} (a_{11} a_{30}-a_{10} a_{31})\right\}\\ +b_3 \left\{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})\right\} \end{array} \right] \end{eqnarray}