<s>[INST] Let \(A\) be an \(n\times n\) matrix. Define a function \(f_A\) by \(f_A(\mathbf{x}) = A\mathbf{x}\), where \(\mathbf{x}\) is an \(n\times 1\) column vector. Suppose that \(f_A\) is one-to-one. Prove that \(A\) must be invertible.[/INST]