定数倍の性質

# 定数倍の性質

$|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ c{a}_{t1}& c{a}_{t2}& \cdots & c{a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$$=c|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

また，この性質は行列式の転置における性質から，ある列の各成分に一定の数 $c$がかけられている場合でも成立する．

## ■具体例

$|\begin{array}{cc}2a& 2b\\ c& d\end{array}|$$=\left(2a\right)d-\left(2b\right)c$$=2\left(ad-bc\right)$$=2|\begin{array}{cc}a& b\\ c& d\end{array}|$

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ 2{a}_{21}& 2{a}_{22}& 2{a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|$

$={a}_{11}\left(2{a}_{22}\right){a}_{33}+{a}_{12}\left(2{a}_{23}\right){a}_{31}$$+{a}_{13}\left(2{a}_{21}\right){a}_{32}-{a}_{11}\left(2{a}_{23}\right){a}_{32}$$-{a}_{12}\left(2{a}_{21}\right){a}_{33}+{a}_{13}\left(2{a}_{22}\right){a}_{31}$

$=2\left({a}_{11}{a}_{22}{a}_{33}+{a}_{12}{a}_{23}{a}_{31}$$+{a}_{13}{a}_{21}{a}_{32}-{a}_{11}{a}_{23}{a}_{32}$$-{a}_{12}{a}_{21}{a}_{33}+{a}_{13}{a}_{22}{a}_{31}\right)$

$=2|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|$

$|\begin{array}{ccc}2& 4& 6\\ 3& 2& 1\\ 4& 5& 6\end{array}|=2|\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\\ 4& 5& 6\end{array}|$

$|\begin{array}{ccc}2& 3& 4\\ 4& 2& 5\\ 6& 1& 6\end{array}|=2|\begin{array}{ccc}1& 3& 4\\ 2& 2& 5\\ 3& 1& 6\end{array}|$

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