Question

# 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]}

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

2020-11-02
The given matrix is
$$\displaystyle{B}={\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$
Reduce the given augmented matrix B in to system of linear equation
Ax=b
The matrix form of the first equation is
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}\\{0}&{0}&{1}&{2}&{0}\\{0}&{0}&{0}&{0}&{1}\end{matrix}\right]}\cdot{\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\\{x}_{{4}}\\{x}_{{5}}\end{matrix}\right]}={\left[\begin{matrix}{0}\\{0}\\{0}\end{matrix}\right]}$$
where
$$\displaystyle{A}={\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}\\{0}&{0}&{1}&{2}&{0}\\{0}&{0}&{0}&{0}&{1}\end{matrix}\right]},{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\\{x}_{{4}}\\{x}_{{5}}\end{matrix}\right]},{b}={\left[\begin{matrix}{0}\\{0}\\{0}\end{matrix}\right]}$$
We use another equation to find a. general solution
$$x_1-x_2-2x_4=0$$
$$x_3+2x_4=0$$
$$x_5=0$$
Then
$$\displaystyle{\left({3}\right)}\Rightarrow{x}_{{1}}={x}_{{2}}+{2}{x}_{{4}}$$
$$\displaystyle{\left({4}\right)}\Rightarrow{x}_{{3}}=-{2}{x}_{{4}}$$
$$\displaystyle{\left({5}\right)}\Rightarrow{x}_{{3}}={0}$$
In vectors form , the general solution , we obtain
$$\displaystyle{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\\{x}_{{4}}\\{x}_{{5}}\end{matrix}\right]}={\left[\begin{matrix}{x}_{{2}}+{2}{x}_{{4}}\\{x}_{{2}}\\{0}\\-{2}{x}_{{4}}\\{0}\end{matrix}\right]}={x}_{{2}}{\left[\begin{matrix}{1}\\{1}\\{0}\\{0}\\{0}\end{matrix}\right]}+{x}_{{4}}{\left[\begin{matrix}{2}\\{0}\\-{2}\\{1}\\{0}\end{matrix}\right]}$$
Hence,
$$\displaystyle{x}={x}_{{2}}{\left[\begin{matrix}{1}\\{1}\\{0}\\{0}\\{0}\end{matrix}\right]}+{x}_{{4}}{\left[\begin{matrix}{2}\\{0}\\-{2}\\{1}\\{0}\end{matrix}\right]}$$
Answer $$\displaystyle{x}={x}_{{2}}{\left[\begin{matrix}{1}\\{1}\\{0}\\{0}\\{0}\end{matrix}\right]}+{x}_{{4}}{\left[\begin{matrix}{2}\\{0}\\-{2}\\{1}\\{0}\end{matrix}\right]}$$