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# On idempotent invertible matrices

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

1. if   then is not necessarily invertible.
2. if then is invertible.
3. is invertible.
4. if then is invertible.

Solution

1. Pick

and note that , and is invertible but is not.

2. 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 .

3. Let , . Since are idempotents we conclude by (ii) that is invertible since is invertible. The result now follows because

4. 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 .

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