Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A^T$ has an inverse. Since $\det(A^T) = \det(A)$, $A^T$ is invertible iff $A$ is invertible. Moreover, $(A^T)^{-1} = (A^{-1})^T$.