Is the following matrix invertible?
$ \begin{bmatrix} x & a & a & \dots & a \\ a & x & a & \dots & a \\ a & a & x & \dots & a \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a & a & a & \dots & x \end{bmatrix} $
I cannot use "determinants" or other stuff but the rank of the matrix. I really tried hard this problem making a lot of algebra and cases but I didn't find the solution; can someone help me for solving it?
Note: The cases when $x=a$ and $a=0$ were covered yet.
Thanks.