# Let A,B in M_n(CC) be matrices and f in C[X] such that Af(B)=B. Prove that if f(B) is not invertible, f(0)=0. Prove that if f(B) is not invertible, f(0)=0.

Let $A,B\in {M}_{n}\left(\mathbb{C}\right)$ be matrices and $f\in \mathbb{C}\left[X\right]$ such that $Af\left(B\right)=B$. Prove that if $f\left(B\right)$ is not invertible, $f\left(0\right)=0$
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Mireya Hoffman
Let $f\left(x\right)=\sum _{k=0}^{n}{a}_{k}{x}^{k}$. Suppose that $f\left(B\right)$ is not invertible, i.e. there exists a nonzero vector v, such that $f\left(B\right)v=0$. Then from $Af\left(B\right)=B$ it follows that $Bv=0$, and therefore
$0=f\left(B\right)v={a}_{0}v+\sum _{k=1}^{n}{a}_{k}{B}^{k}v={a}_{0}v⇒{a}_{0}=0.$