# 行列式の行または列の入れ替えの性質

$|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& & ⋮\\ {a}_{t1}& {a}_{t2}& & {a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& & {a}_{sn}\\ ⋮& ⋮& & ⋮\\ {a}_{11}& {a}_{11}& \cdots & {a}_{11}\end{array}|$$=-|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& & ⋮\\ {a}_{s1}& {a}_{s2}& & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& & {a}_{tn}\\ ⋮& ⋮& & ⋮\\ {a}_{11}& {a}_{11}& \cdots & {a}_{11}\end{array}|$

$t$行と$s$行が入れ替わっている）

また，この性質は行列式の転置における性質から，列どうしの交換でも成立する．

## ■具体例

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

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

$={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}$

$={a}_{22}{a}_{11}{a}_{33}+{a}_{23}{a}_{12}{a}_{31}$$+{a}_{21}{a}_{13}{a}_{32}-{a}_{23}{a}_{11}{a}_{32}$$-{a}_{21}{a}_{12}{a}_{33}-{a}_{22}{a}_{13}{a}_{31}$

$=-{a}_{21}{a}_{12}{a}_{33}-{a}_{22}{a}_{13}{a}_{31}$$-{a}_{23}{a}_{11}{a}_{32}+{a}_{21}{a}_{13}{a}_{32}$$+{a}_{22}{a}_{11}{a}_{33}+{a}_{23}{a}_{12}{a}_{31}$

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

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

$|\begin{array}{ccccc}1& 2& 3& 4& 5\\ {6}& {7}& {8}& {9}& {10}\\ 11& 12& 13& 14& 15\\ {16}& {17}& {18}& {19}& {20}\\ 21& 22& 23& 24& 25\end{array}|$$=-|\begin{array}{ccccc}1& 2& 3& 4& 5\\ {16}& {17}& {18}& {19}& {20}\\ 11& 12& 13& 14& 15\\ {6}& {7}& {8}& {9}& {10}\\ 21& 22& 23& 24& 25\end{array}|$

$|\begin{array}{ccccc}1& {2}& 3& {4}& 5\\ 6& {7}& 8& {9}& 10\\ 11& {12}& 13& {14}& 15\\ 16& {17}& 18& {19}& 20\\ 21& {22}& 23& {24}& 25\end{array}|$$=-|\begin{array}{ccccc}1& {4}& 3& {2}& 5\\ 6& {9}& 8& {7}& 10\\ 11& {14}& 13& {12}& 15\\ 16& {19}& 18& {17}& 20\\ 21& {24}& 23& {22}& 25\end{array}|$

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