行列式の和の性質

行列式の1つの行（または列）の各成分が2つの数の和であるならば， その行（または列）を一方の数のみで置き換えた行列式と，他方のみで置き換えた行列式との和になる（多重線形性）． 

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

⇒証明へ

■具体例

例：2次の行列式の場合

${\displaystyle \left|\begin{array}{cc}{a}_1+a_2& b_1+b_2\\ c& d\end{array}\right|}$  

${\displaystyle =\left(a_1+a_2\right)d-\left(b_1+b_2\right)c}$ ${\displaystyle =a_{1}d+a_{2}d-b_{1}c-b_{2}c}$ ${\displaystyle =a_{1}d-b_{1}c+a_{2}d-b_{2}c}$

${\displaystyle =\left|\begin{array}{cc}{a}_1& b_1\\ c& d\end{array}\right|+\left|\begin{array}{cc}{a}_2& b_2\\ c& d\end{array}\right|}$  

例2：3次の行列式の場合

${\displaystyle \left|\begin{array}{ccc}{a}_1+a_2& b_1+b_2& c_1+c_2\\ d& e& f\\ g& h& i\end{array}\right|}$  

${\displaystyle =\left(a_1+a_2\right)ef+\left(b_1+b_2\right)fg}$${\displaystyle +\left(c_1+c_2\right)dh-\left(a_1+a_2\right)fh}$${\displaystyle +\left(b_1+b_2\right)di+\left(c_1+c_2\right)eg}$

${\displaystyle =a_{1}ef+a_{2}ef+b_{1}fg+b_{2}fg}$ ${\displaystyle +c_{1}dh+c_{2}dh-a_{1}fh-a_{2}fh}$${\displaystyle -b_{1}di-b_{2}di-c_{1}eg-c_{2}eg}$

${\displaystyle =a_{1}ef+b_{1}fg+c_{1}dh-a_{1}fh}$${\displaystyle -b_{1}di-c_{1}eg+a_{2}ef+b_{2}fg}$${\displaystyle +c_{2}dh-a_{2}fh-b_{2}di-c_{2}eg}$

${\displaystyle =\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ d& e& f\\ g& h& i\end{array}\right|+\left|\begin{array}{ccc}{a}_2& b_2& c_2\\ d& e& f\\ g& h& i\end{array}\right|}$  

例3：2次の行列式の変形例

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

例4：3次の行列式の変形例

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

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

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

 

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最終更新日： 2022年8月27日