加加减减之后，矩阵 A 还是原来的“自己”吗？

一、题目

已知 $\boldsymbol{A}=\left[\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}\right]$ 是三阶矩阵, 则下列行列式中等于 $|\boldsymbol{A}|$ 的是哪个？

(A) $\left|\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{1}\right|$

(B) $\left|\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{1}\right|$

(C) $\left|\boldsymbol{\alpha}_{1}+2 \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}\right|$

(D) $\left|\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}\right|$

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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