试确定 $k$ 为何值时, 向量组 $\alpha_1, \alpha_2, \alpha_3$ 的秩为 2 ,
$$
\alpha_1=\left(\begin{array}{c}
1 \\
2 \\
-1 \\
1
\end{array}\right), \alpha_2=\left(\begin{array}{l}
2 \\
0 \\
k \\
0
\end{array}\right) \alpha_3=\left(\begin{array}{c}
0 \\
-4 \\
5 \\
-2
\end{array}\right) .
$$
【答案】 解:
$$
\left(\alpha_1, \alpha_2, \alpha_3\right)=\left(\begin{array}{ccc}
1 & 2 & 0 \\
2 & 0 & -4 \\
-1 & k & 5 \\
1 & 0 & -2
\end{array}\right) \sim\left(\begin{array}{ccc}
1 & 2 & 0 \\
0 & -4 & -4 \\
0 & k+2 & 5 \\
0 & -2 & -2
\end{array}\right) \sim\left(\begin{array}{ccc}
1 & 2 & 0 \\
0 & 1 & 1 \\
0 & 0 & -k+3 \\
0 & 0 & 0
\end{array}\right)
$$
$-k+3=0$, 解得 $k=3$.


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