# 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)

$\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right){x}_{1}+\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right){x}_{2}+\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){x}_{3}={a}_{24}{b}_{1}-{a}_{14}{b}_{2}$　　･･････(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)

$\left({a}_{11}{a}_{34}-{a}_{14}{a}_{31}\right){x}_{1}+\left({a}_{12}{a}_{34}-{a}_{14}{a}_{32}\right){x}_{2}+\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right){x}_{3}={a}_{34}{b}_{1}-{a}_{14}{b}_{3}$　･･････(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)

$\left({a}_{11}{a}_{44}-{a}_{14}{a}_{41}\right){x}_{1}+\left({a}_{12}{a}_{44}-{a}_{14}{a}_{42}\right){x}_{2}+\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right){x}_{3}={a}_{44}{b}_{1}-{a}_{14}{b}_{4}$　･･････(13)

(7)，(10)，(13)より${x}_{3}$を消去する．

(7)×$\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)$

$\begin{array}{l}\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right){x}_{1}+\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right){x}_{2}\\ +\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){x}_{3}=\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{24}{b}_{1}-{a}_{14}{b}_{2}\right)\end{array}$　･･････(14)

(10)×$\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)$

$\begin{array}{c}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{11}{a}_{34}-{a}_{14}{a}_{31}\right){x}_{1}+\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{12}{a}_{34}-{a}_{14}{a}_{32}\right){x}_{2}\\ +\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right){x}_{3}=\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{34}{b}_{1}-{a}_{14}{b}_{3}\right)\end{array}$　･･････(15)

(14)−(15)

$\begin{array}{l}\left\{\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{11}{a}_{34}-{a}_{14}{a}_{31}\right)\right\}{x}_{1}\\ +\left\{\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{12}{a}_{34}-{a}_{14}{a}_{32}\right)\right\}{x}_{2}\\ =\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)\left({a}_{24}{b}_{1}-{a}_{14}{b}_{2}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{34}{b}_{1}-{a}_{14}{b}_{3}\right)\end{array}$　･･････(16)

(16)の${x}_{1}$ の係数を整理する．

$\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right)\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{11}{a}_{34}-{a}_{14}{a}_{31}\right)$

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

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

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

$={a}_{14}\left(-{a}_{11}{a}_{24}{a}_{33}-{a}_{13}{a}_{21}{a}_{34}+{a}_{14}{a}_{21}{a}_{33}+{a}_{13}{a}_{24}{a}_{31}+{a}_{11}{a}_{23}{a}_{34}-{a}_{14}{a}_{23}{a}_{31}\right)$

$={A}_{11}$

とおく．

(16)の${x}_{2}$ の係数を整理する．

$\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right)\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{12}{a}_{34}-{a}_{14}{a}_{32}\right)$

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

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

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

$={a}_{14}\left(-{a}_{12}{a}_{24}{a}_{33}-{a}_{13}{a}_{22}{a}_{34}+{a}_{14}{a}_{22}{a}_{33}+{a}_{13}{a}_{24}{a}_{32}+{a}_{12}{a}_{23}{a}_{34}-{a}_{14}{a}_{23}{a}_{32}\right)$

$={A}_{12}$

とおく．

(16)の定数項を整理する．

$\left({a}_{24}{b}_{1}-{a}_{14}{b}_{2}\right)\left({a}_{13}{a}_{34}-{a}_{14}{a}_{33}\right)-\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{34}{b}_{1}-{a}_{14}{b}_{3}\right)$

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

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

$=-{a}_{14}{a}_{24}{a}_{33}{b}_{1}-{a}_{13}{a}_{14}{a}_{34}{b}_{2}+{a}_{14}{a}_{14}{a}_{33}{b}_{2}+{a}_{13}{a}_{14}{a}_{24}{b}_{3}+{a}_{14}{a}_{23}{a}_{34}{b}_{1}-{a}_{14}{a}_{14}{a}_{23}{b}_{3}$

$={a}_{14}\left(-{a}_{24}{a}_{33}{b}_{1}-{a}_{13}{a}_{34}{b}_{2}+{a}_{14}{a}_{33}{b}_{2}+{a}_{13}{a}_{24}{b}_{3}+{a}_{23}{a}_{34}{b}_{1}-{a}_{14}{a}_{23}{b}_{3}\right)$

$={B}_{1}$

とおく．

(14)−(15)を${A}_{11}{x}_{1}+{A}_{12}{x}_{2}={B}_{1}$ と表す．

(7)×$\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)$

$\begin{array}{l}\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right){x}_{1}+\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right){x}_{2}\\ +\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right){x}_{3}\\ =\left({a}_{24}{b}_{1}-{a}_{14}{b}_{2}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)\end{array}$　･･････(17)

(13)×$\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)$

$\begin{array}{l}\left({a}_{11}{a}_{44}-{a}_{14}{a}_{41}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){x}_{1}+\left({a}_{12}{a}_{44}-{a}_{14}{a}_{42}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){x}_{2}\\ +\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){x}_{3}\\ =\left({a}_{44}{b}_{1}-{a}_{14}{b}_{4}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\end{array}$　･･････(18)

(17)−(18)

$\begin{array}{l}\left\{\left({a}_{11}{a}_{24}-{a}_{14}{a}_{21}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)-\left({a}_{11}{a}_{44}-{a}_{14}{a}_{41}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\right\}{x}_{1}\\ +\left\{\left({a}_{12}{a}_{24}-{a}_{14}{a}_{22}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)-\left({a}_{12}{a}_{44}-{a}_{14}{a}_{42}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\right\}{x}_{2}\\ =\left({a}_{24}{b}_{1}-{a}_{14}{b}_{2}\right)\left({a}_{13}{a}_{44}-{a}_{14}{a}_{43}\right)-\left({a}_{44}{b}_{1}-{a}_{14}{b}_{4}\right)\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\end{array}$　･･････(19)

(19)の ${x}_{1}$ の係数を整理する．

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

(19)の ${x}_{2}$ の係数を整理する．

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

(19)の定数項を整理する．

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

$={B}_{2}$

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)

$\left({A}_{11}{A}_{22}-{A}_{12}{A}_{21}\right){x}_{1}={B}_{1}{A}_{22}-{A}_{12}{B}_{2}$

${A}_{11}{A}_{22}-{A}_{12}{A}_{21}$

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

${={a}_{14}}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{11}{a}_{22}{a}_{33}{a}_{44}-{a}_{11}{a}_{22}{a}_{34}{a}_{43}-{a}_{11}{a}_{23}{a}_{32}{a}_{44}+{a}_{11}{a}_{23}{a}_{34}{a}_{42}\right$

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

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

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

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

$={{a}_{14}}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){C}_{1}$

とおく．

${B}_{1}{A}_{22}-{A}_{12}{B}_{2}$

$\begin{array}{l}=\left\{{a}_{14}\left({b}_{1}{a}_{23}{a}_{34}-{b}_{1}{a}_{24}{a}_{33}+{a}_{13}{a}_{24}{b}_{3}-{a}_{13}{{b}_{2}a}_{34}+{a}_{14}{b}_{2}{a}_{33}-{a}_{14}{a}_{23}{b}_{3}\right)\\ \text{ }×{a}_{14}\left({a}_{12}{a}_{23}{a}_{44}-{a}_{12}{a}_{24}{a}_{43}+{a}_{13}{{a}_{24}a}_{42}-{a}_{13}{a}_{22}{a}_{44}+{a}_{14}{a}_{22}{a}_{43}-{a}_{14}{a}_{23}{a}_{42}\right)\right\}\\ \text{ }-\left\{{a}_{14}\left({a}_{12}{a}_{23}{a}_{34}-{a}_{12}{a}_{24}{a}_{33}+{a}_{13}{a}_{24}{a}_{32}-{a}_{13}{a}_{22}{a}_{34}+{a}_{14}{a}_{22}{a}_{33}-{a}_{14}{a}_{23}{a}_{32}\right)\\ \text{ }×{a}_{14}\left({b}_{1}{a}_{23}{a}_{44}-{b}_{1}{{a}_{24}a}_{43}+{a}_{13}{a}_{24}{b}_{4}-{a}_{13}{{b}_{2}a}_{44}+{a}_{14}{b}_{2}{a}_{43}-{a}_{14}{a}_{23}{b}_{4}\right)}\end{array}$

$={{a}_{14}}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right)\left({a}_{22}{a}_{33}{a}_{44}{b}_{1}-{a}_{22}{a}_{34}{a}_{43}{b}_{1}-{a}_{23}{a}_{32}{a}_{44}{b}_{1}+{a}_{23}{a}_{34}{a}_{42}{b}_{1}\right$

$+{a}_{24}{a}_{32}{a}_{43}{b}_{1}-{a}_{24}{a}_{33}{a}_{42}{b}_{1}-{a}_{12}{a}_{33}{a}_{44}{b}_{2}+{a}_{12}{a}_{34}{a}_{43}{b}_{2}+{a}_{13}{a}_{32}{a}_{44}{b}_{2}$

$-{a}_{13}{a}_{34}{a}_{42}{b}_{2}-{a}_{14}{a}_{32}{a}_{43}{b}_{2}+{a}_{14}{a}_{33}{a}_{42}{b}_{2}+{a}_{12}{a}_{23}{a}_{44}{b}_{3}-{a}_{12}{a}_{24}{a}_{43}{b}_{3}$

$-{a}_{13}{a}_{22}{a}_{44}{b}_{3}+{a}_{13}{a}_{24}{a}_{42}{b}_{3}+{a}_{14}{a}_{22}{a}_{43}{b}_{3}-{a}_{14}{a}_{23}{a}_{42}{b}_{3}-{a}_{12}{a}_{23}{a}_{34}{b}_{4}$

$+{a}_{12}{a}_{24}{a}_{33}{b}_{4}+{a}_{13}{a}_{22}{a}_{34}{b}_{4}-{a}_{13}{a}_{24}{a}_{32}{b}_{4}-{a}_{14}{a}_{22}{a}_{33}{b}_{4}+{a}_{14}{a}_{23}{a}_{32}{b}_{4})$

$={{a}_{14}}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){C}_{2}$

とおく．

${x}_{1}=\frac{{B}_{1}{A}_{22}-{A}_{12}{B}_{2}}{{A}_{11}{A}_{22}-{A}_{12}{A}_{21}}$

$=\frac{{C}_{2}}{{C}_{1}}$

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