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?
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?
Set $f(x)=\det (A-xI)$ where $I$ is the identity matrix.
Show that $f(x)$ is a polynomial of degree $n$ and that $f(A)=0$.