线性无关的矩阵乘以线性无关的矩阵一定得线性无关的矩阵

一、题目

已知 ${\mathit{\alpha }}_1$, ${\mathit{\alpha }}_2$, ${\mathit{\alpha }}_3$ 线性无关，若 ${\mathit{\alpha }}_1-3{\mathit{\alpha }}_3$, $a{\mathit{\alpha }}_1+{\mathit{\alpha }}_2+2{\mathit{\alpha }}_3$, $2{\mathit{\alpha }}_1+3{\mathit{\alpha }}_2+{\mathit{\alpha }}_3$ 亦线性无关，则 $a$ 的取值（）

难度评级：

二、解析

由题可知：

$$({\alpha }_1,{\alpha }_2,{\alpha }_3)\left[\begin{array}{ccc}1& a& 2\\ 0& 1& 3\\ -3& 2& 1\end{array}\right]\Rightarrow \left|\begin{array}{ccc}1& a& 2\\ 0& 1& 3\\ -3& 2& 1\end{array}\right|\ne 0\Rightarrow$$

$$\left|\begin{array}{ccc}1& a& 2\\ 0& 1& 3\\ 0& 2+3a& 7\end{array}\right|\ne 0\Rightarrow$$

$$\left|\begin{array}{ccc}1& a& 2\\ 0& 1& 3\\ 0& 3a& 1\end{array}\right|\ne 0\Rightarrow$$

$$1-9a\ne 0\Rightarrow a\ne \frac{1}{9}$$

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