# 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

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 ${c}_{1}$
$\left\{\begin{array}{l}{x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}=0\\ -{x}_{1}+{x}_{2}-{x}_{3}+{x}_{4}=0\\ {x}_{1}+{x}_{2}-{x}_{3}-{x}_{4}=0\\ 3{x}_{1}+{x}_{2}+{x}_{3}-{x}_{4}=0\end{array}$
$X=\left(\begin{array}{c}1\\ -1\\ -1\\ 1\end{array}\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

d2saint0

Lets write
$A=\left[\begin{array}{cccc}1& 1& 1& 1\\ -1& 1& -1& 1\\ 1& 1& -1& -1\\ 3& 1& 1& -1\end{array}\right]X=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\end{array}\right]$
Matrix A , which has the dimensions $4×4$, is the matrix of ciefficients of the system , X , which has the dimensions $4×1$ is the matrix of unknowns.
$AX=0$
$\left[\begin{array}{cccc}1& 1& 1& 1\\ -1& 1& -1& 1\\ 1& 1& -1& -1\\ 3& 1& 1& -1\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$
If X is a solution of SX=0 , then so is ${c}_{1}$ for any constant ${c}_{1}$ . compute the product AX:
$AX=\left[\begin{array}{cccc}1& 1& 1& 1\\ -1& 1& -1& 1\\ 1& 1& -1& -1\\ 3& 1& 1& -1\end{array}\right]\left[\begin{array}{c}1\\ -1\\ -1\\ 1\end{array}\right]$
$=\left[\begin{array}{c}1\cdot 1+1\cdot \left(-1\right)+1\cdot \left(-1\right)+1\cdot 1\\ -1\cdot 1+1\cdot \left(-1\right)+\left(-1\right)\cdot \left(-1\right)+1\cdot 1\\ 1\cdot 1+1\cdot \left(-1\right)+\left(-1\right)\cdot \left(-1\right)+\left(-1\right)\cdot 1\\ 3\cdot 1+1\cdot \left(-1\right)+1\cdot \left(-1\right)+\left(-1\right)\cdot 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$