= Solution
The finite-dimensional <spectral theorem> says that a <self-adjoint operator> on a complex inner-product space has an <orthonormal basis> of <eigenvectors> and all its <eigenvalues> are real. Equivalently, its matrix is unitarily diagonalisable with real diagonal entries.
For $a\in\mathbb R$, the matrix is equal to its conjugate transpose:
$$
M_a^*=\begin{pmatrix}1&i&a\\-i&1&0\\a&0&1\end{pmatrix}=M_a.
$$
Thus it represents a self-adjoint operator for the standard <inner product> on $\mathbb C^3$. The theorem applies directly, proving
$$
\boxed{M_a\text{ is diagonalisable for every real }a.}
$$
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