How do prove a matrix statement

Question

The problem is pretty simple conceptually but I am not sure how to write proofs out for matrixes.

Answer 1

Theorem: $\det(AB)=\det(A)\det(B)$

It follows from this that if $A^2=A$, then $$\det(A)^2=\det(A)\det(A)=\det(AA)=\det(A^2)=\det(A)$$ so $\det(A)=0$ or $\det(A)=1$.