Let be two idempotent matrices such that is invertible and let . Let be the identity matrix. Show that:

- if then is not necessarily invertible.
- if then is invertible.
- is invertible.
- if then is invertible.

**Solution**

- Pick
and note that , and is invertible but is not.

- It’s clear for . For , suppose that for some . We need to show that . We have thus . But since we also have and hence
because is invertible. So and therefore . So .

- Let , . Since are idempotents we conclude by (ii) that is invertible since is invertible. The result now follows because
- Let and suppose that for some . We are done if we show that . Well, we have and thus implying that because is invertible. Hence which gives because . Thus and therefore .