Solution
= Solution
{c}
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
$$
\det A=0
\iff T_A\text{ is not invertible}
\iff\ker T_A\ne\{0\}
\iff \exists v\ne0:\ Av=0.
$$