The rank and eigenvalues of the operator $T(M) = AM - MA$ on the space of matrices

This problem is from Artin Algebra Second edition, 5.2.3.

Let $A$ be an $n\times n$ complex matrix.

$(a)$ Consider the linear operator $T$ defined on the space $\mathbb{C}^{n\times n}$ of all complex $n\times n$ matrices by the rule $T(M) = AM - MA$. Prove that the rank of this operator is at most $n^2-n$

$(b)$ Determine the eigenvalues of $T$ in terms of the eigenvalues $\lambda_1,\cdots,\lambda_n$ of $A$.

For part $(a)$, I tried to use Dimension Formula. But, I don't know how to show that $\dim(\ker(T))$ is greater than equal to $n$.

For part $(b)$, I really don't know...

Can someone help me?

Hint: if $A$ is diagonal, things are rather simple. Diagonalizable matrices are dense...