题目
14.求解下列齐次线性方程组:(1)}x_(1)+x_(2)+2x_(3)-x_(4)=0,2x_(1)+x_(2)+x_(3)-x_(4)=0,2x_(1)+2x_(2)+x_(3)+2x_(4)=0;(2)}x_(1)+2x_(2)+x_(3)-x_(4)=0,3x_(1)+6x_(2)-x_(3)-3x_(4)=0,5x_(1)+10x_(2)+x_(3)-5x_(4)=0;(3)}2x_(1)+3x_(2)-x_(3)-7x_(4)=0,3x_(1)+x_(2)+2x_(3)-7x_(4)=0,4x_(1)+x_(2)-3x_(3)+6x_(4)=0,x_(1)-2x_(2)+5x_(3)-5x_(4)=0;(4)}3x_(1)+4x_(2)-5x_(3)+7x_(4)=0,2x_(1)-3x_(2)+3x_(3)-2x_(4)=0,4x_(1)+11x_(2)-13x_(3)+16x_(4)=0,7x_(1)-2x_(2)+x_(3)+3x_(4)=0.
14.求解下列齐次线性方程组:
(1)$\begin{cases}x_{1}+x_{2}+2x_{3}-x_{4}=0,\\2x_{1}+x_{2}+x_{3}-x_{4}=0,\\2x_{1}+2x_{2}+x_{3}+2x_{4}=0;\end{cases}$
(2)$\begin{cases}x_{1}+2x_{2}+x_{3}-x_{4}=0,\\3x_{1}+6x_{2}-x_{3}-3x_{4}=0,\\5x_{1}+10x_{2}+x_{3}-5x_{4}=0;\end{cases}$
(3)$\begin{cases}2x_{1}+3x_{2}-x_{3}-7x_{4}=0,\\3x_{1}+x_{2}+2x_{3}-7x_{4}=0,\\4x_{1}+x_{2}-3x_{3}+6x_{4}=0,\\x_{1}-2x_{2}+5x_{3}-5x_{4}=0;\end{cases}$
(4)$\begin{cases}3x_{1}+4x_{2}-5x_{3}+7x_{4}=0,\\2x_{1}-3x_{2}+3x_{3}-2x_{4}=0,\\4x_{1}+11x_{2}-13x_{3}+16x_{4}=0,\\7x_{1}-2x_{2}+x_{3}+3x_{4}=0.\end{cases}$
题目解答
答案
(1) 通解为：
\[
\boxed{
\begin{pmatrix}
-\frac{4}{3} \\
-3 \\
\frac{4}{3} \\
1
\end{pmatrix}k
\]
(2) 通解为：
\[
\boxed{
\begin{pmatrix}
-2 \\
1 \\
0 \\
0 \\
1
\end{pmatrix}k_1 + k_2
\]
(3) 通解为：
\[
\boxed{
\begin{pmatrix}
-\frac{1}{2} \\
\frac{7}{2} \\
\frac{5}{2} \\
1
\end{pmatrix}k
\]
(4) 通解为：
\[
\boxed{
3 \\
19 \\
13 \\
20
\end{pmatrix}k_1 + \begin{pmatrix}
- \frac{13}{17} \\
- \frac{20}{17} \\
1
\end{pmatrix}k_2
\end{pmatrix}
\]
(5) 与原答案一致。