4元1次方程式の解

$a_{11} x_{1} + a_{12} x_{2} + a_{13} x_{3} + a_{14} x_{4} = b_{1}$ 　　･･････(1)

$a_{21} x_{1} + a_{22} x_{2} + a_{23} x_{3} + a_{24} x_{4} = b_{2}$ 　　･･････(2)

$a_{31} x_{1} + a_{32} x_{2} + a_{33} x_{3} + a_{34} x_{4} = b_{3}$ 　　･･････(3)

$a_{41} x_{1} + a_{42} x_{2} + a_{43} x_{3} + a_{44} x_{4} = b_{4}$ 　　･･････(4)

詳しい計算過程はここにあります

$x_{1} = \frac{a_{14}^{2} (a_{13} a_{24} - a_{14} a_{23}) C_{2}}{a_{14}^{2} (a_{13} a_{24} - a_{14} a_{23}) C_{1}}$

$\begin{array}{l} C_{1} = a_{11} a_{22} a_{33} a_{44} - a_{11} a_{22} a_{34} a_{43} - a_{11} a_{23} a_{32} a_{44} + a_{11} a_{23} a_{34} a_{42} \\ + a_{11} a_{24} a_{32} a_{43} - a_{11} a_{24} a_{33} a_{42} - a_{12} a_{21} a_{33} a_{44} + a_{12} a_{21} a_{34} a_{43} \\ + a_{12} a_{23} a_{31} a_{44} - a_{12} a_{23} a_{34} a_{41} - a_{12} a_{24} a_{31} a_{43} + a_{12} a_{24} a_{33} a_{41} \\ + a_{13} a_{21} a_{32} a_{44} - a_{13} a_{21} a_{34} a_{42} - a_{13} a_{22} a_{31} a_{44} + a_{13} a_{22} a_{34} a_{41} \\ + a_{13} a_{24} a_{31} a_{42} - a_{13} a_{24} a_{32} a_{41} - a_{14} a_{21} a_{32} a_{43} + a_{14} a_{21} a_{33} a_{42} \\ + a_{14} a_{22} a_{31} a_{43} - a_{14} a_{22} a_{33} a_{41} - a_{14} a_{23} a_{31} a_{42} + a_{14} a_{23} a_{32} a_{41} \end{array}$

$C_{2} = a_{22} a_{33} a_{44} b_{1} - a_{22} a_{34} a_{43} b_{1} - a_{23} a_{32} a_{44} b_{1} + a_{23} a_{34} a_{42} b_{1} + a_{24} a_{32} a_{43} b_{1} - a_{24} a_{33} a_{42} b_{1} - a_{12} a_{33} a_{44} b_{2} + a_{12} a_{34} a_{43} b_{2} + a_{13} {a_{32} a}_{44} b_{2} - a_{13} {a_{34} a}_{42} b_{2} - a_{14} a_{32} a_{43} b_{2} + a_{14} a_{33} a_{42} b_{2} + a_{12} a_{23} {a44}_{} b_{3} - a_{12} a_{24} a_{43} b_{3} - a_{13} a_{22} a_{44} b_{3} + a_{13} {a_{24} a_{42} b}_{3} + a_{14} a_{22} a_{43} b_{3} - a_{14} a_{23} a_{42} b_{3} - a_{12} a_{23} a_{34} b_{4} + a_{12} a_{24} a_{33} b_{4} + a_{13} a_{22} a_{34} b_{4} - a_{13} a_{24} a_{32} b_{4} - a_{14} a_{22} a_{33} b_{4} + a_{14} a_{23} a_{32} b_{4}$

行列式を用いて表すと

$x_{1} = \frac{{a_{14}}^{2} |\begin{matrix} a_{13} & a_{23} \\ a_{14} & a_{24} \end{matrix}| |\begin{matrix} b_{1} & a_{12} & a_{13} & a_{14} \\ b_{2} & a_{22} & a_{23} & a_{24} \\ b_{3} & a_{32} & a_{33} & a_{34} \\ b_{4} & a_{42} & a_{43} & a_{44} \end{matrix}|}{{a_{14}}^{2} |\begin{matrix} a_{13} & a_{23} \\ a_{14} & a_{24} \end{matrix}| |\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix}|} = \frac{|\begin{matrix} b_{1} & a_{12} & a_{13} & a_{14} \\ b_{2} & a_{22} & a_{23} & a_{24} \\ b_{3} & a_{32} & a_{33} & a_{34} \\ b_{4} & a_{42} & a_{43} & a_{44} \end{matrix}|}{|\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix}|}$

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最終更新日： 2022年10月12日