# Problem of the Week #56 - April 22nd, 2013

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#### Chris L T521

##### Well-known member
Staff member
Thanks again to those who participated in last week's POTW! Here's this week's problem!

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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.

#### Chris L T521

##### Well-known member
Staff member
This week's problem was correctly answered by Sudharaka. You can find his solution below.

Using the Cayley-Hamilton theorem we get,

$p(A)=A^{n} + a_1A^{n-1} + \ldots + a_{n-1}A+a_{n}I_n=0_n$

where $$0_n$$ is the $$n\times n$$ zero matrix.

Since $$A$$ is non-singular multiplying by $$A^{-1}$$ we get,

$A^{n-1} + a_1A^{n-2} + \ldots + a_{n-2}A+a_{n-1}I_n+a_n A^{-1}=0_n$

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

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