If $\begin{bmatrix}XY \\ X\end{bmatrix}$ has full column rank, then does $X$ have full column rank?

Question

Consider the rectangular matrix $X \in \mathbb R^{m \times n}$, the square matrix $Y \in \mathbb R^{n \times n}$, and the block matrix $Z = \begin{bmatrix}XY \\ X\end{bmatrix} \in \mathbb R^{2m \times n}$. If $Z$ has full column rank, then does $X$ necessarily have full column rank as well? I haven't yet been able to think of a counterexample, so I'm interested to know whether this is true or false.

Answer 1

$$ X = \begin{bmatrix} 1 & 0 \end{bmatrix} \\ Y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$

$Z = Y$. $Z$ has full column rank and $X$ does not.

Answer 2

Not true!

Counterexample:

$X=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ which clearly has column rank $1$.

Let $Y=\begin{pmatrix}1&2\\3&4\end{pmatrix}$.

$XY=\begin{pmatrix}1&2\\0&0\end{pmatrix}$

$Z=\begin{pmatrix}1&2\\0&0\\1&0\\0&0\end{pmatrix}$ which has full column rank.