let $A$ belongs to $M_{n \times n}$ then we have to show that there exist a polynomial $f(x)$ with real coefficients such that $f(A)=0$..we know that this is true for characteristic polynomial i.e for every matrix satisfies its characteristic polynomial i mean $f(A)=0$ (by a famous hamilton's theorem..its look like characteristic polynomial is not requirement here due to this reason i really cant understand what the question is??
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Maybe the question preceds Cayley-Hamilton. – Git Gud Mar 22 '14 at 13:32
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sory i couidn't got it?? – hafsah Mar 22 '14 at 13:33
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If you're following a textbook,maybe that problem is supposed to be solved before you learn about Cayley-Hamilton. – Git Gud Mar 22 '14 at 13:34
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nup in text book this problem is asked after cayley hamilton – hafsah Mar 22 '14 at 13:36
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Is $A$ a real matrix? – Git Gud Mar 22 '14 at 13:37
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i think it is..cz polynomial is restricted with real coefficients – hafsah Mar 22 '14 at 13:39
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1I can only think that maybe they want to take $A\in \mathcal M_{n\times n}(\mathbb C)$ and want you to find a real polynomial which annihilates $A$. This isn't simply Cayley-Hamilton. – Git Gud Mar 22 '14 at 13:41
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The 1st $n^2+1$ powers of $A$ live in an $n^2$ dimensional space, hence are linearly dependent.
Gil Bor
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