Product of an invertible matrix and an arbitrary matrix.

Question

Take $B$ to be any $n \times n$ square matrix. If $A$ is an invertible $n \times n$ matrix, then does the set of $AB$ span all $n \times n$ matrices?

Edit: I believe it is true because for a given matrix $C$ you are trying to construct from the product $AB$, you know the entries of $A$ and you know the entries of $C$ so you can construct a system of equations to find the entries of $B$.

Could someone clarify if this reasoning is okay?

Answer 1

Let $C$ be a matrix. Set $B=A^{-1}C$. Then $AB=C$.