Prove that a matrix is not diagonalizable for any scalars $a$, $b$.

Question

I need to prove that \begin{align*} A= \begin{bmatrix} a & 1 & 0\\ 0 & a & 0\\ 0 & 0 & b\\ \end{bmatrix} \end{align*}

is not diagonalizable for any scalars $a$ and $b$.

I've tried just finding the eigenvalues to start with, and I end up with the eigenvalues are $\lambda_1=a$ with multiplicity 2 and $\lambda_2=b$ with mult. 1. Then I find that the dimension of the corresponding eigenspace for $\lambda_1$ has dimension 1 and the corresponding eigenspace for $\lambda_2$ has dimension 1 as well. Since the dimension of the eigenspace of $\lambda_1=a$ doesn't match its multiplicity, then the original matrix is not diagonalizable. Is this correct..?

Answer 1

No, it is wrong. The dimensions of the eigenspaces are $1$, not $2$.