Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 4 / e / Solution

True for these square matrices. The determinant multiplication identity gives $\det (XY)=\det X\det Y$. Since $XY$ is invertible, its determinant is nonzero; both factors are therefore nonzero, so both matrices have matrix inverses.

Equivalently, $Y$ must be injective and $X$ must be surjective, and either property implies invertibility for an endomorphism of a finite-dimensional vector space. The equal square dimensions matter: a rectangular matrix and a right inverse can have identity product without being mutually invertible square matrices.