I've been trying to solve this "simple" problem using the gaussian elimination method, but I don't get the right reduction steps to simplify the matrix and left a simple parameter term in the last row.
So I can't get the matrix rank according to the parameter value.
The matrix is: \begin{pmatrix} 1 & 1 & a & a \\ a & a & 1 & 1 \\ 1 & a & 1 & a \\ \end{pmatrix}
Just any ideas of what reduction steps should I use?, I may solve it using this method, instead of calculating the Det.
Thank you so much.
I assume that the reason you are finding this hard is because the solution does not depend on $a$. Letting $x = -1, y = 1, z = 1$, we see that the all three equations hold regardless of the value of $a$.
So if you were after a value of $a$ that would make this solvable, they all do.
If you were looking for the dependence of $x, y, z$ on $a$, they are all constant.