线性相关的向量组成的行列式一定等于零

题目 01

若向量组 $α_{1} = (a + 1, 1, a)^{⊤}$ , $α_{2} = (a, - 2, 2 - a)^{⊤}$ , $α_{3} = (a - 1, - 3, 4 - a)^{⊤}$ 线性相关, 则 $a = ?$

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解析 01

$| α_{1}, α_{2}, α_{3} | = | \begin{array}{ccc} a + 1 & a & a - 1 \\ 1 & - 2 & - 3 \\ a & 2 - a & 4 - a \end{array} | =$

进行初等行列变换和初等列变换：

$| \begin{array}{ccc} a + 1 & 0 & 0 \\ 1 & - 2 & - 1 \\ 0 & - a & 1 \end{array} | = (a + 1) | \begin{array}{cc} - 2 & - 1 \\ - a & 1 \end{array} | \Rightarrow$

${\begin{cases} a = - 1 \\ a = - 2 \end{cases}$

页码： 12