Eigenvalues of $A$ Let  $A$ be a $3*3$  matrix with real entries. If $A$ commutes with all $3*3$ matrices with real entries, then how many number of distinct real eigenvalues exist for $A$?
please give some hint.
 thank you for your time.

The value $3$ in the question is of no importance (as long as it is not $0$).
If matrices $A,B$ commute, then $A$ must stabilise $\ker B$: if $x\in \ker B$ and $y=A\cdot x$ then $B(y)=B\cdot A(x)=A(B(x))=A(0)=0$ and so $y\in\ker B$.
Now choosing $B$ appropriately $\ker B$ can be any subspace you want, so $A$ stabilises all subspaces. In particular it stabilises $\langle v\rangle$ for any nonzero vector $v$, so $A(v)\in\langle v\rangle$ and $v$ is an eigenvector; in particular (since we are in a real vector space) $A$ has some real eigenvalue$~\lambda$. But then since $A-\lambda I$ commutes with every $B$, the nonzero subspace $\ker(A-\lambda I)$ is a stable by all matrices, which can only mean it is the entire space. So $A=\lambda I$  does not have any other eigenvalues.
