SEEMOUS 2012 Problem 3

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$ , then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$ ?