Show that a linear matrix transformation is bijective iff A is invertible. Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$.
Show that it is bijective IFF $A$ is invertible. 
I was thinking then that I could show that it is surjective. So suppose there exists a 
$B \in  M_n(K)$ such that $T(B) = ?$ What would it equal to show that? 

Hint: Note that a function is bijective if and only if it has a two-sided inverse. Given that $A$ has an inverse $A^{-1}$ such that $A^{-1}A=AA^{-1}=\mathbb 1$, can you come up with an two-sided inverse $T^{-1}: M_n(K) \rightarrow M_n(K)$ for $T$? For the converse, consider $T^{-1}(A)$.
