Let $n \in \mathbb{N}$ with $n \ge 1$, let $k$ be a field, and let $A \in M_n(k)$. Let $T_A:k^n \to k^n$ be the $k$-[linear map](/page/Linear%20Map) defined by $T_A(x)=Ax$. Then the following conditions are equivalent: $A \in GL_n(k)$; $\ker T_A=\{0\}$; $\operatorname{im} T_A=k^n$; $\operatorname{rank} A=n$; and $\det A\ne 0$.