Question

True or false--- (justify *against example*)

If B is an idempotent matrix, then B = In or B is not
invertible.

Answer 1

False

If B is an idempotent matrix, it means that B^2 = B.

However, this statement does not guarantee B is either the identity matrix (In) or not invertible.

Idempotent matrices are a special kind of matrix with special properties but it cannot be used to determine invertibility or whether B=In.

If an idempotent matrix B is given, there is no way to determine whether B is In or invertible from this information alone, as many different matrices can be idempotent.