The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}{0}&{0}&{1}&{2}&{0}&{0}{0}&{0}&{0}&{0}&{1}&{0}end{matrix}right]}

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$\left[\begin{array}{cccccc}1& -1& 0& -2& 0& 0\\ 0& 0& 1& 2& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]$
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The given matrix is
$B=\left[\begin{array}{cccccc}1& -1& 0& -2& 0& 0\\ 0& 0& 1& 2& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]$
Reduce the given augmented matrix B in to system of linear equation
Ax=b
The matrix form of the first equation is
$\left[\begin{array}{ccccc}1& -1& 0& -2& 0\\ 0& 0& 1& 2& 0\\ 0& 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\\ {x}_{5}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
where
$A=\left[\begin{array}{ccccc}1& -1& 0& -2& 0\\ 0& 0& 1& 2& 0\\ 0& 0& 0& 0& 1\end{array}\right],x=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\\ {x}_{5}\end{array}\right],b=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
We use another equation to find a. general solution
${x}_{1}-{x}_{2}-2{x}_{4}=0$
${x}_{3}+2{x}_{4}=0$
${x}_{5}=0$
Then
$\left(3\right)⇒{x}_{1}={x}_{2}+2{x}_{4}$
$\left(4\right)⇒{x}_{3}=-2{x}_{4}$
$\left(5\right)⇒{x}_{3}=0$
In vectors form , the general solution , we obtain
$x=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\\ {x}_{5}\end{array}\right]=\left[\begin{array}{c}{x}_{2}+2{x}_{4}\\ {x}_{2}\\ 0\\ -2{x}_{4}\\ 0\end{array}\right]={x}_{2}\left[\begin{array}{c}1\\ 1\\ 0\\ 0\\ 0\end{array}\right]+{x}_{4}\left[\begin{array}{c}2\\ 0\\ -2\\ 1\\ 0\end{array}\right]$
Hence,
$x={x}_{2}\left[\begin{array}{c}1\\ 1\\ 0\\ 0\\ 0\end{array}\right]+{x}_{4}\left[\begin{array}{c}2\\ 0\\ -2\\ 1\\ 0\end{array}\right]$