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How do I show there is a nonzero polynomial $f$ with $f(A)=0$, where $f(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$ and A is a $n \times n$ matrix?

And isn't that true the set of $f$ with $f(A)=0$ an ideal of the polynomial ring?

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    Yes, it is an ideal; you need to be careful when showing that $(fg)(A) = f(A)g(A)$ (you need to justify it by noting that $A$ commutes with all coefficients and all powers of itself), but otherwise it should not be hard. For the first part, look up the Cayley-Hamilton Theorem.2012-03-28

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Set $f(x)=\det (A-xI)$ where $I$ is the identity matrix.

Show that $f(x)$ is a polynomial of degree $n$ and that $f(A)=0$.