Let $q$ be a prime integer. Let $\mathsf{Y} \in \mathbb{F}_q^{n \times n}$ be a particular full rank matrix.
Consider a matrix $\mathsf{X}$ sampled uniformly at random from the set of all matrices over $\mathbb{F}_q^{n \times n}$.
What is the probability that $\mathsf{X} \mathsf{Y} = \mathsf{Y}$?
Unless $\mathsf{X}$ is the identity matrix, I suspect the probability is going to be negligible. But I could not formalize the intuition.