Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\det(A) \neq 0$. This follows from $\det(AA^{-1}) = \det(I) = 1$, so $\det(A)\det(A^{-1}) = 1$, meaning $\det(A)$ cannot be zero.