For which values of $\ a, T $ is an isomorphism

Question

$\ T: \mathbf R^n \rightarrow \mathbf R^n $, $\ T(v) = Av $

$\ A = \begin{bmatrix} 1 & a & a & \cdots& a \\ a & 1 &a & \cdots & a \\ \vdots & & \ddots & & \vdots \\ \\ a & \cdots & \cdots & \cdots & 1\end{bmatrix} $

for what values of $\ a$ , the transformation will be isomorphism. So looking for cases where $\ \ker T = \{ 0 \} $ , one of them is $\ a = 0 $ but maybe better to try find all cases where $\ T$ isn't isomorphism ?

So Im trying to find $\ Ax = 0 $ where $\ x \not = 0 $ and I can see that $\ a =1 $ satisfies that but I'm not sure what is the way to find all the solutions to this equation? I was thinking of writing the transformation as follows:

$\ T(x_1,x_2,..,x_n) = (x_1 + ax_2+\dots+ax_n, ax_1+x_2+ax_3+\dots+ax_n,\dots,ax_1+\dots+ax_{n-1}+x_n) = \\ (x_1 + a(x_2+\dots+x_n). x_2+ a(x_1+x_3+\dots+x_n), \dots , x_n+a(x_1+\dots+x_{n-1}))
$

Answer 1

Hint:

Try to evaluate $detA$ and find $a:detA\not=0$

Hint for evaluating $detA$:

Add all rows to the first one