Rank of special block matrix

Question

Given two matrices $A\in \mathbb{R}^{a\times b}$ and $X\in \mathbb{R}^{b\times b}$, can there anything be said about the rank of the block matrix:

\begin{align*} \begin{bmatrix} AX & A \end{bmatrix} ~, \end{align*} provided $X$ is non-singular and $a>b$?

Answer 1

This is the rank of $A$, since the columns of $AX$ are linear combinations of the columns of $A$. The premises that $X$ is non-singular and that $a\gt b$ aren't necessary.