Given a SLAE $\left\{\begin{matrix}
5x_{1}+x_{2}-2x_{3}+6x_{4}=0\\
x_{1}+3x_{2}-2x_{3}+4x_{4}=0\\
3x_{1}+2x_{2}-2x_{3}+5x_{4}=0\\
4x_{1}+5x_{2}-4x_{3}+9x_{4}=0
\end{matrix}\right.$
Check if the rows of matrices A and B given below form a fundamental solution of SLAE.
$A=\begin{pmatrix}
4 & 8 & 14 & 0\\
2 & 4 & 7 & 0
\end{pmatrix}$ $\hspace{0.5mm}$ $B=\begin{pmatrix}
-1 & -1 & 0 & 1\\
4 & 8 & 14 & 0
\end{pmatrix}$
I found the row reduced echelon form of the coefficient matrix for the homogeneous SLAE as the following:
$\begin{pmatrix}
1 & 0 & \frac{-2}{7} & 1\\
0 & 1 & \frac{-4}{7} & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{pmatrix}$
But am confused on how to relate it with the matrices A and B?
Can you please let me know the correct approach for the same?
Best Answer
Since the rang of your system is $2$ and the dimension of the space of variables is $2$, solutions of the system form a $4-2 = 2$-dimensional subspace. Fundamental solution is a basis of this subspace. So you should check whether rows of $A$ and $B$ provide solutions to the system or not. Then, in the first case, you should prove or disprove that they are linearly independent. If they actually are, they form a fundamental system by the dimensionality argument.