If AB = A and BA = B, then which of the following is/are true?

Given, AB = A, BA = B⇒ B×AB=B×Aor (BA)B = B or B2 = BAlso,A×BA=AB⇒(AB)A=A⇒A2=ANow AT2=AT×AT=(A×A)T=A2T=ATSimilarly, (BT)2 = BTHence, AT and BT are idempotent.