Let $B$ be a complex $n\times n$ matrix. Prove or disprove: The linear operator $T$ on the space of all $n\times n$ matrices defined by $ T(A) = AB - BA $ is singular.
$T(I)=0$, where $I$ is the identity. So.. the map has a nontrivial kernal.
Hint. Consider rank-nullity. Does the map have a nontrivial kernel? That is, is there a subspace of matrices that commute with $B$? (In fact, there is a nontrivial subspace of matrices that commute with every complex $n\times n$ matrix...)
Well, if $B$ is the zero matrix, this is simple, and if not, what is $T(B)$?