Answer
$\mathbf{A}$ must have 5 rows and 4 columns.
Work Step by Step
If $T(\vec{x})=\mathbf{A}\vec{x}=\vec{b}$ where $\vec{x}$ is an 4-dimensional vector and $\vec{b}$ is a 5-dimensional vector, then then $\mathbf{A}$ must have 5 rows and 4 columns. This follows from the way matrix multiplication has been defined.
