Suppose there are two matrices as follows:
$M=[A~B]$, $N=[C~B]$,
where $A\in\mathbb{R}^{2\times 4}$, $B\in\mathbb{R}^{2\times 6}$, $C\in\mathbb{R}^{2\times 2}$.
Suppose Im$(A)=$Im$(C)$, where Im$(X)$ denotes the image (column space) of $X$.
My question is “Is the rank of $M$ the same as the rank of $N$?” (In case the rank of $B$ is not full rank.)
Yes. Let The rank of $M$ is the dimension of the span of the columns of $A$ and the columns of $B$. The rank of $N$ is the dimension of the span of the columns of $C$ and the colmuns of $B$. Since the columns of $A$ and the columns of $C$ span the same subspace by assumption, we are done.