how to prove this Theorem null matrix?

Question

Theorem: The only idempotent matrix whose eigenvalues are all zero is the null matrix.

Then how to prove this?

Answer 1

Suppose $A$ is idempotent, and $A\ne 0$.

Let $x$ be such that $Ax\ne 0$, and let $y=Ax$.

Then $Ay= A(Ax) = A^2x = Ax = y$, so $y$ is an eigenvector with eigenvalue $1$.

Answer 2

If all eigenvalues are zero then $T^n = 0$. Since $T=T^2 = \cdots = T^n$ then we see that $T=0$.