System of Linear Equations - how many solutions? For which real values of t does the following system of linear equations:
$$
\left\{ 
\begin{array}{c}
tx_1 + x_2 + x_3 = 1 \\ 
x_1 + tx_2 + x_3 = 1 \\ 
x_1 + x_2 + tx_3 = 1
\end{array}
\right. 
$$
Have:  a) a unique solution? 
b) infinitely many solutions? 
c) no solutions?
I haven't done linear algebra in almost a year, so I'm really rusty and could use some pushes in the right direction.

$\text{I have used the Gauss elimination and then studied the rank of the coefficient matrix:}$
$\text{1)If}\ t=1 \text{ the system reduces to just one equation, and it has}\ \infty^2 \text{solutions.}$
$\text{2)If }\ t=-2 \text{ there are no solutions.}$
$\text{3)If}\ t≠1,-2 \text{ there is a unique solution, depending on t. }$
