Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\text{nullity}(A) = 0$. By the rank-nullity theorem, $\text{rank}(A) + \text{nullity}(A) = n$, so nullity 0 means rank $n$.