Can an idempotent matrix only have eigenvalue $1$?

Question

A matrix $A\in M_n(\Bbb R)$ is idempotent if $A^2=A$. And it is also easy to show that every idempotent matrix has eigenvalues at most $0$ and $1$. However, I want to find an example that $A$ is idempotent, but $A$ only have eigenvalue $1$. Is it possible?

Answer 1

As Hans Lundmark pointed out, the identity matrix $I$ is an example of such a matrix. In fact this is the only possibility: if $A$ doesn't have an eigenvalue $0$, then $A$ is invertible, and multiplying both sides of $A^2 = A$ by $A^{-1}$ gives $A = I$.