Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.

Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = b$ has a unique solution for every $b \in \mathbb{R}^n$ (or $\mathbb{C}^n$).

Proof: If $A$ is nonsingular, $x = A^{-1}b$ is the unique solution. Conversely, if a unique solution exists for every $b$, then in particular for each standard basis vector $e_j$, there is a unique $x_j$ such that $Ax_j = e_j$. The matrix with columns $x_1, \ldots, x_n$ is $A^{-1}$.