TITLE: Is $T$ singular or nonsingular for $T(A)=AB - BA$?
QUESTION [5 upvotes]: 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.

REPLY [15 votes]: $T(I)=0$, where $I$ is the identity. So.. the map has a nontrivial kernal.

REPLY [4 votes]: Well, if $B$ is the zero matrix, this is simple, and if not, what is $T(B)$?