Chapter 4 - Eigenvalues and Eigenvectors - 4.2 Determinants - Exercises 4.2 - Page 282: 55

$det(A)=0$ or $det(A)=1$

We know that if $A$ and $B$ are $n\times n$ matrices, then $det(AB)=det(A)det(B)$. If $A^2=A$, then $det(A^2)=det(A)\\det(A^2)-det(A)=0\\det(A)det(A)-det(A)=0\\det(A)(det(A)-1)=0$ Thus $det(A)=0$ or $det(A)=1$