Question

If A and B are two square matrices of order 3 $\times$ 3 which satisfy AB = A and BA = B then $(A+B{)}^7$ is

• A. $7(A+B)$
• B. ${7.1}_{3\times 3}$
• C. $64(A+B)$
• D. ${1281}_{3\times 3}$

Answer 1

The correct option is C $64(A+B)$

AB =A, BA = B $\Rightarrow A^2=A$ and $B^2=B$
$(A+B{)}^2=A^2+B^2+AB+BA$
= A+B+A+B = 2(A+B)
$(A+B{)}^3=(A+B{)}^2(A+B)=2(A+B{)}^2=2^2(A+B)\therefore (A+B{)}^7=2^6(A+B)=64(A+B)$