Let \(A\) be an \(n \times n\) matrix over a field \(F\). Show that the set of all polynomials \(f\) in \(F[x]\) such that \(f(A) = 0\) is an ideal.
Solution
Let \[ I=\{f\in F[x]:f(A)=0\}. \] Certainly \(0\in I\). If \(f,g\in I\), then \[ (f-g)(A)=f(A)-g(A)=0, \] so \(f-g\in I\). If \(h\in F[x]\) and \(f\in I\), then \[ (hf)(A)=h(A)f(A)=0, \] so \(hf\in I\). Therefore \(I\) is an ideal of \(F[x]\).