Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ represents the identity map with respect to some pair of bases. The identity map is invertible, so its matrix representation must be nonsingular.
Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ represents the identity map with respect to some pair of bases. The identity map is invertible, so its matrix representation must be nonsingular.