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Let

p(Y)=(-Y)^n + a_(n-1)*Y^(n-1) + ... + a_0

be the characteristic polynomial of matrix A. Show that A is invertible, if and only if a_0 isn't zero and that inverse of A is

A^(-1)=q(A),

where q is polynomial.


So i'm having trouble showing that inverse is polynomial of A.

For the first part i went this way; A is invertible when its determinant isnt zero. p(Y)=det(A-YI), so p(0)=det(A), so det(A)=a_0.

For inverse i dont even know how to start. Any guidance is much appreciated!

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Hint: Multiply Cayley-Hamilton relation $$ A^n + a_{n-1} A^{n-1} + \cdots + a_1 A + a_0 = 0$$ by $A^{-1}.$

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  • $\begingroup$ Oh, thats so simple, thank you very much! $\endgroup$
    – leia
    Sep 9 '14 at 16:56
  • $\begingroup$ @leia you are welcome. good luck. $\endgroup$
    – ir7
    Sep 9 '14 at 16:57
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some more hints can be found in Cayley-Hamilton theorem, http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem

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