# 行列式の和の性質

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

## ■具体例

$|\begin{array}{cc}{a}_{1}+{a}_{2}& {b}_{1}+{b}_{2}\\ c& d\end{array}|$

$=\left({a}_{1}+{a}_{2}\right)d-\left({b}_{1}+{b}_{2}\right)c$ $={a}_{1}d+{a}_{2}d-{b}_{1}c-{b}_{2}c$ $={a}_{1}d-{b}_{1}c+{a}_{2}d-{b}_{2}c$

$=|\begin{array}{cc}{a}_{1}& {b}_{1}\\ c& d\end{array}|+|\begin{array}{cc}{a}_{2}& {b}_{2}\\ c& d\end{array}|$

$|\begin{array}{ccc}{a}_{1}+{a}_{2}& {b}_{1}+{b}_{2}& {c}_{1}+{c}_{2}\\ d& e& f\\ g& h& i\end{array}|$

$=\left({a}_{1}+{a}_{2}\right)ef+\left({b}_{1}+{b}_{2}\right)fg$$+\left({c}_{1}+{c}_{2}\right)dh-\left({a}_{1}+{a}_{2}\right)fh$$+\left({b}_{1}+{b}_{2}\right)di+\left({c}_{1}+{c}_{2}\right)eg$

$={a}_{1}ef+{a}_{2}ef+{b}_{1}fg+{b}_{2}fg$ $+{c}_{1}dh+{c}_{2}dh-{a}_{1}fh-{a}_{2}fh$$-{b}_{1}di-{b}_{2}di-{c}_{1}eg-{c}_{2}eg$

$={a}_{1}ef+{b}_{1}fg+{c}_{1}dh-{a}_{1}fh$$-{b}_{1}di-{c}_{1}eg+{a}_{2}ef+{b}_{2}fg$$+{c}_{2}dh-{a}_{2}fh-{b}_{2}di-{c}_{2}eg$

$=|\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ d& e& f\\ g& h& i\end{array}|+|\begin{array}{ccc}{a}_{2}& {b}_{2}& {c}_{2}\\ d& e& f\\ g& h& i\end{array}|$

$|\begin{array}{cc}1& 2\\ 3& 4\end{array}|=|\begin{array}{cc}1+0& 0+2\\ 3& 4\end{array}|$$=|\begin{array}{cc}1& 0\\ 3& 4\end{array}|+|\begin{array}{cc}0& 2\\ 3& 4\end{array}|$

$|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}|$

$=|\begin{array}{ccc}1+0& 0+2& 0+3\\ 4& 5& 6\\ 7& 8& 9\end{array}|$$=|\begin{array}{ccc}1& 0& 0\\ 4& 5& 6\\ 7& 8& 9\end{array}|+|\begin{array}{ccc}0& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}|$$=|\begin{array}{ccc}1& 0& 0\\ 4& 5& 6\\ 7& 8& 9\end{array}|+|\begin{array}{ccc}0+0& 2+0& 0+3\\ 4& 5& 6\\ 7& 8& 9\end{array}|$

$=|\begin{array}{ccc}1& 0& 0\\ 4& 5& 6\\ 7& 8& 9\end{array}|+|\begin{array}{ccc}0& 2& 0\\ 4& 5& 6\\ 7& 8& 9\end{array}|+|\begin{array}{ccc}0& 0& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}|$

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