Given a matrix, is there always another matrix which commutes with it? Given a matrix $A$ over a field $F$, does there always exist a matrix $B$ such that $AB = BA$? (except the trivial case and the polynomial ring?)

Another  example is the adjoint of $A$:
$$
A \operatorname{adj}(A)= \operatorname{adj}(A) A = \det(A)I
$$
(but for invertible matrices it is equal to the scalar $\det(A)$ multipliying the inverse of $A$, so is trivial that commutes. with $A$).
