Problem of finding values of $ a $ for which two matrices are similar Problem: Let $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ 1 & 2 & 7-a^{2} \\ 2 & 2+a & 6\end{array}\right) $ , $ B=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 9\end{array}\right)$ where $ a \in \mathbb{R} $. Find all the values of $ a $ for which $ A $ is similar to $ B $
Attempt:
We know the necessary conditions for two matrices to be similar are:
Their traces are equal
They have the same determinant
They have the same characteristic polynomial
They have the same eigenvalues
Their ranks are equal
Thus, $ |A| = [ 12 - (7-a^2)(2+a) ] - 2[ 6-2(7-a^2)] + 3[ 2+a - 4 ] =  (2-a)^2\cdot (2+a) $,
$ |B| = 0 $, hence $ |A| = 0 $ and therefore either $ a= 2 $ or $ a = -2 $
The characteristic polynomial of $ B $ is $ \Delta_B(x) = x^2(9-x) $
How do I continue from here? I'm stuck. ( I thought to jordanize $ A $ and $ B $ but that seems to really complicate things [ jordanizing $ A $ seems really hard here ], I also thought I'd find the eigenspaces of the eigenvalues of $ B $ but I still wasen't sure how I'd continue from there )
Thanks in advance for help!

Hint: $A$ is similar to $B$ .
$B$ is diagonal matrix.
Hence, $A$ is diagonalizable.
And Eigenvaules of $A$ are $ 0 $ and $9$ and they are of same multiplicity as of multiplicity of $0$ and $9$ of $B$
