Question: Suppose that \(A \in M_{n}\) is nonsingular. According to \((1.1 .7)\), this is equivalent to assuming that \(0 \notin \sigma(A) .\) For each \(\lambda \in \sigma(A)\), show that \(\lambda^{-1} \in \sigma\left(A^{-1}\right) .\) If \(A x=\lambda x\) and \(x \neq 0\), show that \(A^{-1} x=\lambda^{-1} x\).

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