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.

If $A$ commute with $B$ then $$[A,B] = AB - BA = 0$$
$$AB=BA$$ 
$$ABB^{-1}=BAB^{-1}$$ 
$$A=BAB^{-1}$$
If the $B$ has inverse matrix $B^{-1}$, then sets of eigenvalues not change.
Matrix $3\times{3}$ has 3 eigenvalues.
