rank and orthogonal projection

Question

If $A$ is an orthogonal projection matrix onto a subspace $W$ of dimension $2$ in $\mathbb{R}^4$,

how can I prove that $${\rm rank}(A) = 2$$ $${\rm rank}(A^2) = 2$$ $${\rm rank}(A^2) = 1$$ $${\rm rank}(A^2) = 4$$

Answer 1

Let $x$ be any vector in $\mathbb{R}^4$. Then $Ax \in W$ by definition of projection, so since $W$ is two-dimensional, it follows that $\text{rank}(A) = 2$. Also, by the definition of projection, $A^2 = A$. So for the same reasons as before, $\text{rank}(A^2) = 2$. Orthogonality of $A$ is not needed.