Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = b$ has at least one solution for every $b \in \mathbb{R}^n$.
Proof: If $A$ is nonsingular, $x = A^{-1}b$ is a solution. Conversely, if solutions exist for all $b$, the columns of $A$ span $\mathbb{R}^n$. Since there are $n$ columns in an $n$-dimensional space, they must be a basis, hence $A$ is nonsingular.
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