Chapter 7 - Eigenvalues and Eigenvectors - 7.1 Eigenvalues and Eigenvectors - 7.1 Exercises - Page 352: 77

Either $1$ or $0$.

Let $A$ be an idempotent matrix. This means that $A^2-A=0$, so the minimal polynomial of $A$ is $m(x)=x^2-x$. The possible eigenvalues of the matrix $A$ are the zeroes of the minimal polynomials. Since the zeroes of $m(x)$ are $\{0,1\}$, the possible eigenvalues are $0$ or $1$.