# 行列式の計算則

$|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}+c{a}_{s1}& {a}_{t2}+c{a}_{s2}& \cdots & {a}_{tn}+c{a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$$=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

## ■証明

$|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}+c{a}_{s1}& {a}_{t2}+c{a}_{s2}& \cdots & {a}_{tn}+c{a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

$=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {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}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

$=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {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}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

$=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|+0$

$=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{s1}& {a}_{s2}& \cdots & {a}_{sn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{t1}& {a}_{t2}& \cdots & {a}_{tn}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}|$

## ■具体例

$\left|A\right|=\left|\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 3& 4& 5& 1\\ 3& 4& 5& 1& 2\\ 4& 5& 1& 2& 3\\ 5& 1& 2& 3& 4\end{array}\right|$

### 例1

$\left|\begin{array}{ccccc}{1}& {2}& {3}& {4}& {5}\\ 2& 3& 4& 5& 1\\ {3}& {4}& {5}& {1}& {2}\\ 4& 5& 1& 2& 3\\ 5& 1& 2& 3& 4\end{array}\right|$$=\left|\begin{array}{ccccc}{1}& {2}& {3}& {4}& {5}\\ 2& 3& 4& 5& 1\\ {3}+{1}×{2}& {4}+{2}×{2}& {5}+{3}×{2}& {1}+{4}×{2}& {2}+{5}×{2}\\ 4& 5& 1& 2& 3\\ 5& 1& 2& 3& 4\end{array}\right|$

### 例2

$\left|\begin{array}{ccccc}\hfill {1}\hfill & \hfill 2\hfill & \hfill {3}\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill {2}\hfill & \hfill 3\hfill & \hfill {4}\hfill & \hfill 5\hfill & \hfill 1\hfill \\ \hfill {3}\hfill & \hfill 4\hfill & \hfill {5}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill {4}\hfill & \hfill 5\hfill & \hfill {1}\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill {5}\hfill & \hfill 1\hfill & \hfill {2}\hfill & \hfill 3\hfill & \hfill 4\hfill \end{array}\right|$$=\left|\begin{array}{ccccc}\hfill {1}\hfill & \hfill 2\hfill & \hfill {3}+{1}×{3}\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill {2}\hfill & \hfill 3\hfill & \hfill {4}+{2}×{3}\hfill & \hfill 5\hfill & \hfill 1\hfill \\ \hfill {3}\hfill & \hfill 4\hfill & \hfill {5}+{3}×{3}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill {4}\hfill & \hfill 5\hfill & \hfill {1}+{4}×{3}\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill {5}\hfill & \hfill 1\hfill & \hfill {2}+{5}×{3}\hfill & \hfill 3\hfill & \hfill 4\hfill \end{array}\right|$

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