# 係数行列

$x$$y$を未知数とする$2$$1$次連立方程式

$\left\{\begin{array}{c}2x+4y=14\\ 3x-2y=5\end{array}$

$x$$y$の係数を成分とする行列

$\left(\begin{array}{cc}2& 4\\ 3& -2\end{array}\right)$

のことを係数行列という． 参照：拡大係数行列

$2$$1$次連立方程式

$\left\{\begin{array}{c}2x+4y=14\\ 3x-2y=5\end{array}$

で表された関数を列ベクトルを使って表すと

$\left(\begin{array}{c}2x+4y\\ 3x-2y\end{array}\right)=\left(\begin{array}{c}14\\ 5\end{array}\right)$

となる．

$\left(\begin{array}{c}2x+4y\\ 3x-2y\end{array}\right)=\left(\begin{array}{cc}2& 4\\ 3& -2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

すなわち，連立方程式の係数行列 $\left(\begin{array}{cc}2& 4\\ 3& -2\end{array}\right)$ と未知数を成分とした列ベクトル$\left(\begin{array}{c}x\\ y\end{array}\right)$ の積となる．

$\left(\begin{array}{cc}2& 4\\ 3& -2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}14\\ 5\end{array}\right)$

と表される．

$\left\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}={b}_{2}\\ ⋮\\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\cdots +{a}_{m}{x}_{n}={b}_{m}\end{array}$

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m1}& {a}_{m2}& \cdots & {a}_{mn}\end{array}\right)$

となる．

$\left\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}={b}_{2}\\ ⋮\\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\cdots +{a}_{m}{x}_{n}={b}_{m}\end{array}$

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m1}& {a}_{m2}& \cdots & {a}_{mn}\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)=\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{m}\end{array}\right)$

となる．

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m1}& {a}_{m2}& \cdots & {a}_{mn}\end{array}\right)=A$$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)=x$$\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{m}\end{array}\right)=b$

とおくと，連立方程式は

$Ax=b$

と表せる．

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