题目
21.(本题满分12分)已知向量组alpha_(1)=(}10-1-1).(1)证明:alpha_(1),alpha_(2)是alpha_(1),alpha_(2),alpha_(3),alpha_(4)的极大线性无关组;(2)求矩阵H使得A=GH,并求A^10.
21.(本题满分12分)
已知向量组$\alpha_{1}=\left(\begin{matrix}1\\0\\-1\\-1\end{matrix}\right)$,$\alpha_{2}=\left(\begin{matrix}1\\-1\\0\\-2\end{matrix}\right)$,$\alpha_{3}=\left(\begin{matrix}0\\-1\\1\\-1\end{matrix}\right)$,$\alpha_{4}=\left(\begin{matrix}0\\1\\-1\\1\end{matrix}\right)$.设$A=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$, $G=(\alpha_{1},\alpha_{2})$.
(1)证明:$\alpha_{1},\alpha_{2}$是$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$的极大线性无关组;
(2)求矩阵H使得A=GH,并求$A^{10}$.
题目解答
答案
(1) $\alpha_1, \alpha_2$ 线性无关，且 $\alpha_3, \alpha_4$ 可由其线性表示，故为极大线性无关组。
(2) $H = \begin{pmatrix} 1 & 0 & -1 & 1 \\ 0 & 1 & 1 & -1 \end{pmatrix}$，
$A^{10} = G(HG)^9H = \boxed{
\begin{pmatrix}
59049 & 59049 & -59049 & 59049 \\
0 & 0 & 0 & 0 \\
-59049 & -59049 & 59049 & -59049 \\
-59049 & -59049 & 59049 & -59049
\end{pmatrix}
}$。