Find Rank of given Linear transformation.

Question

Suppose $Q \in M_{3 \times 3}\mathbb(R)$ is a matrix of rank $2$.

Let $T : M_{3 \times 3}\mathbb(R) \to M_{3 \times 3}\mathbb(R)$ be the linear transformation defined by $T(P) = QP$. Then rank of T is:

I feel that rankT = rank Q = $2$. But I don't know how to prove it.

Any idea on how to solve it ?

Thank you.

Answer 1

Notice that $QP=0$ if and only if $\operatorname{range}P\subseteq \ker Q$, therefore $$\operatorname{rk}T=9-\dim\ker T=9-\dim\hom\left(\Bbb R^3,\ker Q\right)=9-3\times 1=6$$