Suppose $ A $ be a real matrix such that $ text {trace} (A) = text {trace} (A ^ 2) = text {trace} (A ^ 3) $. What can we conclude about the proper values of A?

My attempts:

- Considered $ A $ as a diagonal matrix and $ lambda_1, lambda_2

, ldots, lambda_n $ Be your own values So we get $$ lambda_1 + lambda_2 + cdots lambda_n = lambda_1 ^ 2 + lambda_2 ^ 2 + cdots + lambda_n ^ 2 = lambda_1 ^ 3 + lambda_2 ^ 3 + cdots + lambda_n ^ 3

From the first equality and using the Cauchy-Schwarz inequality, we could obtain $ lambda_1 + lambda_2 + cdots lambda_n leq n $. My guess is that all eigenvalues must be 1 (just an intuition). But we can not move forward. - Suppose $ A $ is a $ 2 times 2 $ matrix, then we could get trace ($ A ^ 2 $) = (trace ($ A)) ^ 2-2 det (A) $ and trace ($ A ^ 3 $) = (trace ($ A)) ^ 3- (3 det (A) cdot text {trace} (A)) $. Here, too, we could not find a way.

Thanks in advance.