加加减减之后，矩阵 A 还是原来的“自己”吗？

一、题目

已知 $\mathit{A}=[{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3]$ 是三阶矩阵, 则下列行列式中等于 $|\mathit{A}|$ 的是哪个？

(A) $|{\mathit{\alpha }}_1-{\mathit{\alpha }}_2,{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_3-{\mathit{\alpha }}_1|$

(B) $|{\mathit{\alpha }}_1+{\mathit{\alpha }}_2,{\mathit{\alpha }}_2+{\mathit{\alpha }}_3,{\mathit{\alpha }}_3+{\mathit{\alpha }}_1|$

(C) $|{\mathit{\alpha }}_1+2{\mathit{\alpha }}_2,{\mathit{\alpha }}_3,{\mathit{\alpha }}_1+{\mathit{\alpha }}_2|$

(D) $|{\mathit{\alpha }}_1,{\mathit{\alpha }}_2+{\mathit{\alpha }}_3,{\mathit{\alpha }}_1+{\mathit{\alpha }}_2|$

难度评级：

二、解析

A:

$$\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right|A\Rightarrow \left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right|=1-1=0$$

B:

$$\left|\begin{array}{lll}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right|A\Rightarrow \left|\begin{array}{lll}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right|=1+1=2$$

C:

$$\left|\begin{array}{lll}1& 2& 0\\ 0& 0& 1\\ 1& 1& 0\end{array}\right|A\Rightarrow \left|\begin{array}{lll}1& 2& 0\\ 0& 0& 1\\ 1& 1& 0\end{array}\right|=2-1=1$$

D:

$$\left|\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 1& 0\end{array}\right|A\Rightarrow \left|\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 1& 0\end{array}\right|=-1$$

综上可知，正确选项为 C.

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