Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 1 / a / ii / Solution

When $A$ is an invertible matrix, $B=A^{-1}(AB)$. Apply i first to $AB$ and then to $A^{-1}(AB)$:Thus the two row spaces are equal. In particular, invertible left matrix multiplication preserves row rank.