What values of $a$ make matrix $A$ diagonalisable? I have the following question in an assignment paper.

Let $$A=\begin{bmatrix} 0 & a & 0\\ 1 & 0 & a\\ a & 1 & 0\end{bmatrix}$$ For what values of $a$ is $A$ diagonalisable?

Simply put, I don't know how to do it. In the $2 \times 2$ case we were asked, I completed the square of the characteristic polynomial and found that in all but $1$ choice of the unknown entry you got distinct eigenvalues and, therefore, distinct eigenvectors. At which point I just had to consider the one case for which I had eigenvalue of algebraic multiplicity $2$ and show that the geometric multiplicity of the eigenvector associated with it was $1$, I was done. 
Any tips would be hugely appreciated, I've said it an assignment so reservation on full solution I understand but some hints would be amazing. Thank you.

Taking $a^3 = \frac{32}{27} \; , $ this includes complex $a:$
$$
\frac{1}{864a}
\left(
\begin{array}{rrr}
40 & 36 a^2 & 24 a \\
-100 & -90 a^2 & 156 a \\
-9 a^2 & -24 a & 32
\end{array}
\right)
\left(
\begin{array}{rrr}
0 & a & 0 \\
1 & 0 & a \\
a & 1 & 0
\end{array}
\right)
\left(
\begin{array}{rrr}
6 a & -12 a & 54 a^2 \\
9 a^2 & 9 a^2 & -60 \\
10 & 4 & 0
\end{array}
\right) =
\left(
\begin{array}{ccc}
\frac{3}{2}  a^2 & 0 & 0 \\
0 & - \frac{3}{4} a^2 & 1 \\
0 & 0 & - \frac{3}{4} a^2
\end{array}
\right)
$$
