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- Jan 26, 2012

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**Problem**: Let $A$ be an $n\times n$ matrix whose characteristic polynomial is

\[p(\lambda)=\lambda^n+a_1\lambda^{n-1}+\ldots+ a_{n-1}\lambda+ a_n.\]

If $A$ is nonsingular, show that

\[A^{-1}=-\frac{1}{a_n}\left( A^{n-1} + a_1A^{n-2} + \ldots + a_{n-2}A+a_{n-1}I_n\right).\]

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**Hint**:

Use the Cayley-Hamilton theorem.

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