Let $A$ and $B$ be two $3 \times 3$ invertible matrices such that $A$ is an idempotent matrix. Then find $\det B$.
$\mathrm{adj}(B)B=\det(B)I$, so $\det(\mathrm{adj}(B))\det(B)=\det(B)^3$. Since $\det(\mathrm{adj}(B))=1$ we get $\det(B)=\det(B)^3$, so $\det(B)^2=1$. It follows that $\det(B)=\pm1$.