4元1次方程式の計算

■4元1次方程式

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

$x_{4}$ を消去する．

(1)× $a_{24}$

$a_{11} a_{24} x_{1} + a_{12} a_{24} x_{2} + a_{13} a_{24} x_{3} + a_{14} a_{24} x_{4} = a_{24} b_{1}$ 　　･･････(5)

(2)× $a_{14}$

$a_{14} a_{21} x_{1} + a_{14} a_{22} x_{2} + a_{14} a_{23} x_{3} + a_{14} a_{24} x_{4} = a_{14} b_{2}$ 　　･･････(6)

(5)－(6)

$(a_{11} a_{24} - a_{14} a_{21}) x_{1} + (a_{12} a_{24} - a_{14} a_{22}) x_{2} + (a_{13} a_{24} - a_{14} a_{23}) x_{3} = a_{24} b_{1} - a_{14} b_{2}$ 　　･･････(7)

(1)× $a_{34}$

$a_{11} a_{34} x_{1} + a_{12} a_{34} x_{2} + a_{13} a_{34} x_{3} + a_{14} a_{34} x_{4} = a_{34} b_{1}$ 　　･･････(8)

(3)× $a_{14}$

$a_{14} a_{31} x_{1} + a_{14} a_{32} x_{2} + a_{14} a_{33} x_{3} + a_{14} a_{34} x_{4} = a_{14} b_{3}$ 　･･････(8)

(8)－(9)

$(a_{11} a_{34} - a_{14} a_{31}) x_{1} + (a_{12} a_{34} - a_{14} a_{32}) x_{2} + (a_{13} a_{34} - a_{14} a_{33}) x_{3} = a_{34} b_{1} - a_{14} b_{3}$ 　･･････(10)

(1)× $a_{44}$

$a_{11} a_{44} x_{1} + a_{12} a_{44} x_{2} + a_{13} a_{44} x_{3} + a_{14} a_{44} x_{4} = a_{44} b_{1}$ 　･･････(11)

(4)× $a_{14}$

$a_{14} a_{41} x_{1} + a_{14} a_{42} x_{2} + a_{14} a_{43} x_{3} + a_{14} a_{44} x_{4} = a_{14} b_{4}$ 　･･････(12)

(11)－(12)

$(a_{11} a_{44} - a_{14} a_{41}) x_{1} + (a_{12} a_{44} - a_{14} a_{42}) x_{2} + (a_{13} a_{44} - a_{14} a_{43}) x_{3} = a_{44} b_{1} - a_{14} b_{4}$ 　･･････(13)

(7)，(10)，(13)より $x_{3}$ を消去する．

(7)× $(a_{13} a_{34} - a_{14} a_{33})$

$\begin{array}{l} (a_{13} a_{34} - a_{14} a_{33}) (a_{11} a_{24} - a_{14} a_{21}) x_{1} + (a_{13} a_{34} - a_{14} a_{33}) (a_{12} a_{24} - a_{14} a_{22}) x_{2} \\ + (a_{13} a_{34} - a_{14} a_{33}) (a_{13} a_{24} - a_{14} a_{23}) x_{3} = (a_{13} a_{34} - a_{14} a_{33}) (a_{24} b_{1} - a_{14} b_{2}) \end{array}$ 　･･････(14)

(10)× $(a_{13} a_{24} - a_{14} a_{23})$

$\begin{matrix} (a_{13} a_{24} - a_{14} a_{23}) (a_{11} a_{34} - a_{14} a_{31}) x_{1} + (a_{13} a_{24} - a_{14} a_{23}) (a_{12} a_{34} - a_{14} a_{32}) x_{2} \\ + (a_{13} a_{24} - a_{14} a_{23}) (a_{13} a_{34} - a_{14} a_{33}) x_{3} = (a_{13} a_{24} - a_{14} a_{23}) (a_{34} b_{1} - a_{14} b_{3}) \end{matrix}$ 　･･････(15)

(14)－(15)

$\begin{array}{l} {(a_{13} a_{34} - a_{14} a_{33}) (a_{11} a_{24} - a_{14} a_{21}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{11} a_{34} - a_{14} a_{31})} x_{1} \\ + {(a_{13} a_{34} - a_{14} a_{33}) (a_{12} a_{24} - a_{14} a_{22}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{12} a_{34} - a_{14} a_{32})} x_{2} \\ = (a_{13} a_{34} - a_{14} a_{33}) (a_{24} b_{1} - a_{14} b_{2}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{34} b_{1} - a_{14} b_{3}) \end{array}$ 　･･････(16)

(16)の $x_{1}$ の係数を整理する．

$(a_{11} a_{24} - a_{14} a_{21}) (a_{13} a_{34} - a_{14} a_{33}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{11} a_{34} - a_{14} a_{31})$

$\begin{array}{l} = a_{11} a_{24} a_{13} a_{34} - a_{11} a_{24} a_{14} a_{33} - a_{14} a_{21} a_{13} a_{34} + a_{14} a_{21} a_{14} a_{33} \\ - a_{13} a_{24} a_{11} a_{34} + a_{13} a_{24} a_{14} a_{31} + a_{14} a_{23} a_{11} a_{34} - a_{14} a_{23} a_{14} a_{31} \end{array}$

$\begin{array}{l} = a_{11} a_{13} a_{24} a_{34} - a_{11} a_{14} a_{24} a_{33} - a_{13} a_{14} a_{21} a_{34} + a_{14} a_{14} a_{21} a_{33} \\ - a_{11} a_{13} a_{24} a_{34} + a_{13} a_{14} a_{24} a_{31} + a_{11} a_{14} a_{23} a_{34} - a_{14} a_{14} a_{23} a_{31} \end{array}$

$= - a_{11} a_{14} a_{24} a_{33} - a_{13} a_{14} a_{21} a_{34} + a_{14} a_{14} a_{21} a_{33} + a_{13} a_{14} a_{24} a_{31} + a_{11} a_{14} a_{23} a_{34} - a_{14} a_{14} a_{23} a_{31}$

$= a_{14} (- a_{11} a_{24} a_{33} - a_{13} a_{21} a_{34} + a_{14} a_{21} a_{33} + a_{13} a_{24} a_{31} + a_{11} a_{23} a_{34} - a_{14} a_{23} a_{31})$

$= A_{11}$

とおく．

(16)の $x_{2}$ の係数を整理する．

$(a_{12} a_{24} - a_{14} a_{22}) (a_{13} a_{34} - a_{14} a_{33}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{12} a_{34} - a_{14} a_{32})$

$\begin{array}{l} = a_{12} a_{24} a_{13} a_{34} - a_{12} a_{24} a_{14} a_{33} - a_{14} a_{22} a_{13} a_{34} + a_{14} a_{22} a_{14} a_{33} \\ - a_{13} a_{24} a_{12} a_{34} + a_{13} a_{24} a_{14} a_{32} + a_{14} a_{23} a_{12} a_{34} - a_{14} a_{23} a_{14} a_{32} \end{array}$

$\begin{array}{l} = a_{12} a_{13} a_{24} a_{34} - a_{12} a_{14} a_{24} a_{33} - a_{13} a_{14} a_{22} a_{34} + a_{14} a_{14} a_{22} a_{33} \\ - a_{12} a_{13} a_{24} a_{34} + a_{13} a_{14} a_{24} a_{32} + a_{12} a_{14} a_{23} a_{34} - a_{14} a_{14} a_{23} a_{32} \end{array}$

$= - a_{12} a_{14} a_{24} a_{33} - a_{13} a_{14} a_{22} a_{34} + a_{14} a_{14} a_{22} a_{33} + a_{13} a_{14} a_{24} a_{32} + a_{12} a_{14} a_{23} a_{34} - a_{14} a_{14} a_{23} a_{32}$

$= a_{14} (- a_{12} a_{24} a_{33} - a_{13} a_{22} a_{34} + a_{14} a_{22} a_{33} + a_{13} a_{24} a_{32} + a_{12} a_{23} a_{34} - a_{14} a_{23} a_{32})$

$= A_{12}$

とおく．

(16)の定数項を整理する．

$(a_{24} b_{1} - a_{14} b_{2}) (a_{13} a_{34} - a_{14} a_{33}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{34} b_{1} - a_{14} b_{3})$

$\begin{array}{l} = a_{24} b_{1} a_{13} a_{34} - a_{24} b_{1} a_{14} a_{33} - a_{14} b_{2} a_{13} a_{34} + a_{14} b_{2} a_{14} a_{33} \\ - a_{13} a_{24} a_{34} b_{1} + a_{13} a_{24} a_{14} b_{3} + a_{14} a_{23} a_{34} b_{1} - a_{14} a_{23} a_{14} b_{3} \end{array}$

$\begin{array}{l} = a_{13} a_{24} a_{34} b_{1} - a_{14} a_{24} a_{33} b_{1} - a_{13} a_{14} a_{34} b_{2} + a_{14} a_{14} a_{33} b_{2} \\ - a_{13} a_{24} a_{34} b_{1} + a_{13} a_{14} a_{24} b_{3} + a_{14} a_{23} a_{34} b_{1} - a_{14} a_{14} a_{23} b_{3} \end{array}$

$= - a_{14} a_{24} a_{33} b_{1} - a_{13} a_{14} a_{34} b_{2} + a_{14} a_{14} a_{33} b_{2} + a_{13} a_{14} a_{24} b_{3} + a_{14} a_{23} a_{34} b_{1} - a_{14} a_{14} a_{23} b_{3}$

$= a_{14} (- a_{24} a_{33} b_{1} - a_{13} a_{34} b_{2} + a_{14} a_{33} b_{2} + a_{13} a_{24} b_{3} + a_{23} a_{34} b_{1} - a_{14} a_{23} b_{3})$

$= B_{1}$

とおく．

(14)－(15)を $A_{11} x_{1} + A_{12} x_{2} = B_{1}$ と表す．

(7)× $(a_{13} a_{44} - a_{14} a_{43})$

$\begin{array}{l} (a_{11} a_{24} - a_{14} a_{21}) (a_{13} a_{44} - a_{14} a_{43}) x_{1} + (a_{12} a_{24} - a_{14} a_{22}) (a_{13} a_{44} - a_{14} a_{43}) x_{2} \\ + (a_{13} a_{24} - a_{14} a_{23}) (a_{13} a_{44} - a_{14} a_{43}) x_{3} \\ = (a_{24} b_{1} - a_{14} b_{2}) (a_{13} a_{44} - a_{14} a_{43}) \end{array}$ 　･･････(17)

(13)× $(a_{13} a_{24} - a_{14} a_{23})$

$\begin{array}{l} (a_{11} a_{44} - a_{14} a_{41}) (a_{13} a_{24} - a_{14} a_{23}) x_{1} + (a_{12} a_{44} - a_{14} a_{42}) (a_{13} a_{24} - a_{14} a_{23}) x_{2} \\ + (a_{13} a_{44} - a_{14} a_{43}) (a_{13} a_{24} - a_{14} a_{23}) x_{3} \\ = (a_{44} b_{1} - a_{14} b_{4}) (a_{13} a_{24} - a_{14} a_{23}) \end{array}$ 　･･････(18)

(17)－(18)

$\begin{array}{l} {(a_{11} a_{24} - a_{14} a_{21}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{11} a_{44} - a_{14} a_{41}) (a_{13} a_{24} - a_{14} a_{23})} x_{1} \\ + {(a_{12} a_{24} - a_{14} a_{22}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{12} a_{44} - a_{14} a_{42}) (a_{13} a_{24} - a_{14} a_{23})} x_{2} \\ = (a_{24} b_{1} - a_{14} b_{2}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{44} b_{1} - a_{14} b_{4}) (a_{13} a_{24} - a_{14} a_{23}) \end{array}$ 　･･････(19)

(19)の $x_{1}$ の係数を整理する．

$\begin{array}{l} (a_{11} a_{24} - a_{14} a_{21}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{11} a_{44} - a_{14} a_{41}) \\ = a_{11} a_{24} a_{13} a_{44} - a_{11} a_{24} a_{14} a_{43} - a_{14} a_{21} a_{13} a_{44} + a_{14} a_{21} a_{14} a_{43} \\ - a_{13} a_{24} a_{11} a_{44} + a_{13} a_{24} a_{14} a_{41} + a_{14} a_{23} a_{11} a_{44} - a_{14} a_{23} a_{14} a_{41} \\ = - a_{11} a_{14} a_{24} a_{43} - a_{13} a_{14} a_{21} a_{44} + a_{14} a_{14} a_{21} a_{43} \\ + a_{13} a_{14} a_{24} a_{41} + a_{11} a_{14} a_{23} a_{44} - a_{14} a_{14} a_{23} a_{41} \\ = a_{14} (- a_{11} a_{24} a_{43} - a_{13} a_{21} a_{44} + a_{14} a_{21} a_{43} + a_{13} {a_{24} a}_{41} + a_{11} a_{23} a_{44} - a_{14} a_{23} a_{41}) \\ = A_{21} \end{array}$

(19)の $x_{2}$ の係数を整理する．

$\begin{array}{l} (a_{12} a_{24} - a_{14} a_{22}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{13} a_{24} - a_{14} a_{23}) (a_{12} a_{44} - a_{14} a_{42}) \\ = a_{12} a_{24} a_{13} a_{44} - a_{12} a_{24} a_{14} a_{43} - a_{14} a_{22} a_{13} a_{44} + a_{14} a_{22} a_{14} a_{43} \\ - a_{13} a_{24} a_{12} a_{44} + a_{13} a_{24} a_{14} a_{42} + a_{14} a_{23} a_{12} a_{44} - a_{14} a_{23} a_{14} a_{42} \\ = - a_{12} a_{14} a_{24} a_{43} - a_{13} a_{14} a_{22} a_{44} + a_{14} a_{14} a_{22} a_{43} \\ + a_{13} a_{14} a_{24} a_{42} + a_{12} a_{14} a_{23} a_{44} - a_{14} a_{14} a_{23} a_{42} \\ = a_{14} (- a_{12} a_{24} a_{43} - a_{13} a_{22} a_{44} + a_{14} a_{22} a_{43} + a_{13} {a_{24} a}_{42} + a_{12} a_{23} a_{44} - a_{14} a_{23} a_{42}) \\ = A_{22} \end{array}$

(19)の定数項を整理する．

$\begin{array}{l} (a_{24} b_{1} - a_{14} b_{2}) (a_{13} a_{44} - a_{14} a_{43}) - (a_{44} b_{1} - a_{14} b_{4}) (a_{13} a_{24} - a_{14} a_{23}) \\ = a_{24} b_{1} a_{13} a_{44} - a_{24} b_{1} a_{14} a_{43} - a_{14} b_{2} a_{13} a_{44} + a_{14} b_{2} a_{14} a_{43} \\ - a_{44} b_{1} a_{13} a_{24} + a_{44} b_{1} a_{14} a_{23} + a_{14} b_{4} a_{13} a_{24} - a_{14} b_{4} a_{14} a_{23} \\ = - a_{14} {a_{24} a}_{43} b_{1} - a_{13} a_{14} a_{44} b_{2} + a_{14} a_{14} a_{43} b_{2} \\ + a_{14} a_{23} a_{44} b_{1} + a_{13} {a_{14} a}_{24} b_{4} - a_{14} a_{14} a_{23} b_{4} \\ = a_{14} (- {a_{24} a}_{43} b_{1} - a_{13} a_{44} b_{2} + a_{14} a_{43} b_{2} + a_{23} a_{44} b_{1} + a_{13} a_{24} b_{4} - a_{14} a_{23} b_{4}) \end{array}$

$= B_{2}$

とおく．

(17)－(18)を

$A_{21} x_{1} + A_{22} x_{2} = B_{2}$

と表す．

$A_{11} x_{1} + A_{12} x_{2} = B_{1}$ 　･･････(20)

$A_{21} x_{1} + A_{22} x_{2} = B_{2}$ 　･･････(21)

$X$ を消去する．

(20)× $A_{22}$

$A_{11} A_{22} x_{1} + A_{12} A_{22} x_{2} = B_{1} A_{22}$ 　･･････(22)

(21)× $A_{12}$

$A_{12} A_{21} x_{1} + A_{12} A_{22} x_{2} = A_{12} B_{2}$ 　･･････(23)

(22)－(23)

$(A_{11} A_{22} - A_{12} A_{21}) x_{1} = B_{1} A_{22} - A_{12} B_{2}$

$A_{11} A_{22} - A_{12} A_{21}$

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

${= a_{14}}^{2} (a_{13} a_{24} - a_{14} a_{23}) (a_{11} a_{22} a_{33} a_{44} - a_{11} a_{22} a_{34} a_{43} - a_{11} a_{23} a_{32} a_{44} + a_{11} a_{23} a_{34} a_{42}$

$+ a_{11} a_{24} a_{32} a_{43} - a_{11} a_{24} a_{33} a_{42} - a_{12} a_{21} a_{33} a_{44} + a_{12} a_{21} a_{34} a_{43} + a_{12} a_{23} a_{31} a_{44}$

$- a_{12} a_{23} a_{34} a_{41} - a_{12} a_{24} a_{31} a_{43} + a_{12} a_{24} a_{33} a_{41} + a_{13} a_{21} a_{32} a_{44} - a_{13} a_{21} a_{34} a_{42}$

$- a_{13} a_{22} a_{31} a_{44} + a_{13} a_{22} a_{34} a_{41} + a_{13} a_{24} a_{31} a_{42} - a_{13} a_{24} a_{32} a_{41} - a_{14} a_{21} a_{32} a_{43}$

$+ a_{14} a_{21} a_{33} a_{42} + a_{14} a_{22} a_{31} a_{43} - a_{14} a_{22} a_{33} a_{41} - a_{14} a_{23} a_{31} a_{42} + a_{14} a_{23} a_{32} a_{41})$

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

とおく．

$B_{1} A_{22} - A_{12} B_{2}$

$\begin{array}{l} = {a_{14} (b_{1} a_{23} a_{34} - b_{1} a_{24} a_{33} + a_{13} a_{24} b_{3} - a_{13} {b_{2} a}_{34} + a_{14} b_{2} a_{33} - a_{14} a_{23} b_{3}) \\ \times a_{14} (a_{12} a_{23} a_{44} - a_{12} a_{24} a_{43} + a_{13} {a_{24} a}_{42} - a_{13} a_{22} a_{44} + a_{14} a_{22} a_{43} - a_{14} a_{23} a_{42})} \\ - {a_{14} (a_{12} a_{23} a_{34} - a_{12} a_{24} a_{33} + a_{13} a_{24} a_{32} - a_{13} a_{22} a_{34} + a_{14} a_{22} a_{33} - a_{14} a_{23} a_{32}) \\ \times a_{14} (b_{1} a_{23} a_{44} - b_{1} {a_{24} a}_{43} + a_{13} a_{24} b_{4} - a_{13} {b_{2} a}_{44} + a_{14} b_{2} a_{43} - a_{14} a_{23} b_{4})\} \end{array}$

$= {a_{14}}^{2} (a_{13} a_{24} - a_{14} a_{23}) (a_{22} a_{33} a_{44} b_{1} - a_{22} a_{34} a_{43} b_{1} - a_{23} a_{32} a_{44} b_{1} + a_{23} a_{34} a_{42} b_{1}$

$+ a_{24} a_{32} a_{43} b_{1} - a_{24} a_{33} a_{42} b_{1} - a_{12} a_{33} a_{44} b_{2} + a_{12} a_{34} a_{43} b_{2} + a_{13} a_{32} a_{44} b_{2}$

$- a_{13} a_{34} a_{42} b_{2} - a_{14} a_{32} a_{43} b_{2} + a_{14} a_{33} a_{42} b_{2} + a_{12} a_{23} a_{44} b_{3} - a_{12} a_{24} a_{43} b_{3}$

$- a_{13} a_{22} a_{44} b_{3} + a_{13} a_{24} a_{42} b_{3} + a_{14} a_{22} a_{43} b_{3} - a_{14} a_{23} a_{42} b_{3} - a_{12} a_{23} a_{34} b_{4}$

$+ a_{12} a_{24} a_{33} b_{4} + a_{13} a_{22} a_{34} b_{4} - a_{13} a_{24} a_{32} b_{4} - a_{14} a_{22} a_{33} b_{4} + a_{14} a_{23} a_{32} b_{4})$

$= {a_{14}}^{2} (a_{13} a_{24} - a_{14} a_{23}) C_{2}$

とおく．

$x_{1} = \frac{B_{1} A_{22} - A_{12} B_{2}}{A_{11} A_{22} - A_{12} A_{21}}$

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

$= \frac{C_{2}}{C_{1}}$

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最終更新日： 2022年10月18日