$tr(A)=tr(A^2)=tr(A^3) \implies A$ is Nilpotent

Question

Let $A$ be a square matrix with $\mathrm{tr}(A)=\mathrm{tr}(A^2)=\mathrm{tr}(A^3)$. Prove that $A$ is nilpotent.

The tip in the question is to find the matrix $B$ which is similar to $A$ and then $A^2 \sim B^2$.

Answer 1

As trace of a matrix is equal to the sum of eigevalues $\lambda_i$ and we know that eigenvalues of a matrix powers are equal to powers of eigenvalues from conditions of a task we can obtain

$\sum \lambda_i=\sum \lambda_i^2=\sum \lambda_i^3$

Such conditions are satisfied for $\lambda_i=0$ or $\lambda_i=1$.
Nilpotent matrix has only eigenvalues $0$ hence we see that broader class of matrices can satisfy conditions shown in the question.