# 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]}

Question
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 $$\displaystyle{x}={x}_{{2}}{\left[\begin{matrix}{1}\\{1}\\{0}\\{0}\\{0}\end{matrix}\right]}+{x}_{{4}}{\left[\begin{matrix}{2}\\{0}\\-{2}\\{1}\\{0}\end{matrix}\right]}$$

### Relevant Questions

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system \lambda_1=3 \Rightarrow \left\{ \begin{bmatrix}1\\1\\0 \end{bmatrix} \right\} , \lambda_2=0 \Rightarrow \left\{ \begin{bmatrix}1\\5\\1 \end{bmatrix} , \begin{bmatrix}2\\1\\4 \end{bmatrix} \right\}
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=2i \Rightarrow \left\{ \begin{bmatrix}1+i\\ 2-i \end{bmatrix} \right\} , \lambda_2=-2i \Rightarrow \left\{ \begin{bmatrix}1-i\\ 2+i \end{bmatrix} \right\}$$
The coefficient matrix for a system of linear differential equations of the form $$y_1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$
The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}$$
$$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations $$\begin{cases}9x-3y+z=13 \\ 12x-8z=5 \\ 3x+4y-z =6 \end{cases}$$