# 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&00&1&2&3&0end{bmatrix}

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. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$

2021-02-09
The given matrix is
$$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$
Since A is in reduced echelon form , we find the general solution
$$x_1-x_3-2x_4=0$$
$$x_2+2x_3+3x_4=0$$
Then
$$(1) \Rightarrow x_1=x_3+2x_4$$
$$(2) \Rightarrow x_2=-2x_3-3x_4$$
In vectors form , the general solution , we obtain
$$x= \begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4 \end{bmatrix} =\begin{bmatrix}x_3+2x_4\\ -2x_3-3x_4\\ x_3\\ x_4 \end{bmatrix}=\begin{bmatrix}x_3\\ -2x_3\\ x_3\\ 0 \end{bmatrix}+ \begin{bmatrix}2x_4\\ -3x_4\\ 0\\ x_4 \end{bmatrix}=x_3\begin{bmatrix}1\\ -2\\1\\ 0 \end{bmatrix}+x_4\begin{bmatrix}2\\ -3\\ 0\\1 \end{bmatrix}$$
Hence ,
$$x=x_3\begin{bmatrix}1\\ -2\\1\\ 0 \end{bmatrix}+x_4\begin{bmatrix}2\\ -3\\ 0\\1 \end{bmatrix}$$

### Relevant Questions

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

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Demonstrate that the system of equations is inconsistent.

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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
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$
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. $$\begin{bmatrix}1&-2&0&0&-3\\0&0&1&0&-4\\0&0&0&1&5\end{bmatrix}$$
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 $$y^1=Ay$$  has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda1=3\Rightarrow \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$$
$$\lambda2=0\Rightarrow \begin{bmatrix} 1 \\ 5 \\ 1 \end{bmatrix}\begin{bmatrix}2 \\ 1 \\ 4 \end{bmatrix}$$
$$\begin{bmatrix}1&0&-1&3\\0&1&2&5\\0&0&0&0\end{bmatrix}$$