Recall that a matrix is said to be **idempotent** if

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

i) if then is not necessarily invertible

ii) if then is invertible

iii) is invertible

iv) if then is invertible.

**Solution**. i) Choose

See that and is invertible but is not invertible.

ii) It’s clear for For suppose that for some We need to show that Well, we have and thus But since we also have and hence

because is invertible. So and therefore So

iii) Let See that are idempotents and so, since is invertible, is invertible, by ii). The result now follows because

iv) 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