Compute the determinant The following problem is taken from here exercise $2:$

Question: Evaluate the determinant: 
  \begin{vmatrix}
0 & x & x & \dots & x \\
y & 0 & x & \dots & x \\
y & y & 0 & \dots & x \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
y & y & y & \dots & 0
\end{vmatrix}

My attempt: 
I tried to use first row substract second row to obtain 
\begin{pmatrix}
y & -x & 0 \dots & 0
\end{pmatrix}
and also first row subtracts remaining rows. 
However, I have no idea how to proceed. 

If you continue subtracting the $(k+1)$th row from the $k$th, you end up with
$$ \Delta_n = \begin{vmatrix} -y & x  \\
& -y & x \\
&& -y & x \\
&&&\ddots & \ddots \\
&&&& -y & x \\
y & y & y & \cdots & y & 0 \end{vmatrix}, $$
where blank spaces are zeros. One can also pull out a factor of $xy$ now, but there's not a lot of point. Expanding along the first column gives
$$ \Delta_n = -y\Delta_{n-1} +(-1)^{n-1} y \begin{vmatrix}  x  \\
-y & x \\
& -y & x \\
&&\ddots & \ddots \\
&&& -y & x \end{vmatrix} = -y \Delta_{n-1} + (-1)^{n-1} yx^{n-1}. $$
Iterating this and using $\Delta_2=-xy$ gives
$$ \Delta_n = (-1)^{n-1} (xy^{n-1}+x^2y^{n-2}+\dotsb+x^{n-1}y) = (-1)^{n-1}xy \frac{x^{n-1}-y^{n-1}}{x-y}.  $$
