Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma function.

What use might it be to take the factorial of a matrix? Do any applications come to mind, or does this – for now* – seem to be restricted to the domain of recreational mathematics?

^{(*Until e.g. theoretical physics turns out to have a use for this, as happened with Calabi–Yau manifolds and superstring theory...)}