find a value in order matrix has solution

Question

$\begin{bmatrix}1&0&1\\1&1&a\\1&0&1+a\\1&-a&1\end{bmatrix}\begin{bmatrix} x\\ y\\z\\\end{bmatrix}=\begin{bmatrix} 1\\ a+1\\1\\a+1\end{bmatrix}$

find a in order the solution exist and find the solution.

$\begin{bmatrix}1&0&1&1\\0&1&a-1&a\\0&0&a&0\\0&-a&0&a\end{bmatrix}$ , R2-R1,R3-R1,R4-R1

$\begin{bmatrix}1&0&1&1\\0&1&a-1&a\\0&0&a&0\\0&0&a(a-1)&a(a+1)\end{bmatrix}$ , R4+a.R2

is this mean $a=0$ is the only answer ? thanks!

Answer 1

It means that whenever $a\neq1$ there’s a solution. (In case that $a=0$ there are infinitely many ones.)