Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 3 / a / i / Solution

For the linear map $T_A:x\mapsto Ax$ on the finite-dimensional vector space ${\mathbb{R}}^n$, the rank-nullity theorem implies that injectivity and surjectivity are equivalent. Consequently $T_A$ is invertible if and only if $\ker T_A=\{0\}$.

Using the permitted equivalence between invertibility and a nonzero determinant, we obtain