Exercise2.5
  1. Which is the only matrix that is both a projector and involutory?
  2. Which is the only matrix that is both idempotent and nilpotent?
  3. Prove that if \(A\) is a projector, then so too is \(I-A\text{.}\)
  4. Prove that if \(A\) and \(B\) are both projectors, and \(AB=BA\text{,}\) them \(AB\) is also a projector.
  5. Prove that \(A\) is involutory if and only if \((I-A)(I+A)=0\text{.}\)
  6. Show that if \(A\) is involutory \(B=\frac{1}{2}(I+A)\text{,}\) then \(B\) is a projector.
in-context