# 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=03x_1+x_2+x_3-x_4=0 end{cases} X =begin{pmatrix}1-1-11end{pmatrix}

Question
Forms of linear equations
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 $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$

2021-03-05
Lets write
$$A=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} X=\begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}$$
Matrix A , which has the dimensions 4 x 4 , is the matrix of ciefficients of the system , X , which has the dimensions 4 x1 is the matrix of unknowns.
$$AX=0$$
$$\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}=\begin{bmatrix}0\\ 0\\0\\ 0\end{bmatrix}$$
If X is a solution of SX=0 , then so is c_1X for any constant c_1 . compute the product AX:
$$AX=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}1\\ -1\\ -1\\1\end{bmatrix}$$
$$=\begin{bmatrix}1\cdot1+1\cdot(-1)+1\cdot(-1)+1\cdot1\\-1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+1\cdot1\\1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+(-1)\cdot1\\3\cdot1+1\cdot(-1)+1\cdot(-1)+(-1)\cdot1\end{bmatrix}= \begin{bmatrix}0\\0\\ 0\\0\end{bmatrix}$$

### Relevant Questions

Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If $$\displaystyle{x}_{{h}}$$ is a solution of the system Ax=0, then $$\displaystyle{x}_{{1}}+{x}_{{h}}$$ is a solution of the system Ax=b.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is a solution of the system Ax=0
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
(b) Determine whether the system is consistent. If it is consistent, give the solution.
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$
Write the vector form of the general solution of the given system of linear equations.
$$x_1+2x_2-x_3=0$$
$$x_1+x_2+x_3=0$$
$$x_1+3x_2-3x_3=0$$
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
$$\begin{cases}x_1+x_2-3x_3=-1\\-x_1+2x_2=1\\x_1-x_2+x_3=2\end{cases}$$

Let B be a $$(4\times3)(4\times3)$$ matrix in reduced echelon form.

a) If B has three nonzero rows, then determine the form of B.

b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a $$(4\times3)(4\times3)$$ matrix row equivalent to B.

Demonstrate that the system of equations is inconsistent.

Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$$x=at+b , y=ct+d , z=et+f$$
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2-x_3=1$$
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio $$x_1$$ (in$) and the amount spent advertising in the newspaper $$x_2$$ (in $) according to $$y=ax_1+bx_2+c$$ The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. $$\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline 2400 & { 800} & { 36,000} \\ \hline 2000 & { 500} & { 30,000} \\ \hline 3000 & { 1000} & { 44,000} \\ \hline\end{array}$$ a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $$y=ax_1+bx_2+c$$ d) Predict the monthly sales if the grocer spends$250 advertising on the radio and \$500 advertising in the newspaper for a given month.
$$3x_1+x_2-x_3+x_4=0$$
$$2x_1+2x_2+4x_3-6x_4=0$$
$$2x_1+x_2+3x_3-x_4=0$$
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$