Question

6. If \( A \) and \( B \) are square matrices such that \( AB = B \) and \( BA = A \), then which of the following is/are always true ? (A) \( A \) is an idempotent matrix (B) \( B \) is an involutary matrix (C) \( A^{6}+B^{6}=A^{8}+B^{8} \) (D) \( A^{2}+B^{2}=A+1 \)

Answer 1

Given, AB=B......(i)

           BA=A......(ii)

Post multiply eq(i) with A

=>ABA=BA

=>A²=A (from eqii).....(iii)

Hence A is idempotent matrix

Similarly B²=B.....(iv)

From(iii) A⁸=A &A⁶=A....(v)

From iv,B⁸=B & B⁶= B....(vi)

From (v)&(vi), A⁸+B⁸ = A⁶+B⁶