Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ has a right inverse: there exists $C$ such that $AC = I$. For square matrices, right inverse implies left inverse.
Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ has a right inverse: there exists $C$ such that $AC = I$. For square matrices, right inverse implies left inverse.