Q: If $B$ is a square matrix such that $B^3= B$ then what can we say about the eigenvalues of $B$? Suppose $B$ has eigenvalue $\lambda$ then $B^3$ has eigenvalue $\lambda^3$, Since $B^3=B$ they have the same eigenvalues $\implies \lambda^3 = \lambda  \implies \lambda = 1,0 $
So I could conclude that the eigenvalues of $B$ are either 0 or 1. Is there anything else I can say about $B$? or rather, is my above argument watertight? 

A: You can say $x^3-x$ is a multiple of the minimal polynomial of $B$. As this polynomial has only simple roots, the same is true for the minimal polynomial, hence $B$ is diagonalisable.
