Let `A`, `B` are square matrices of same order satisfying `AB=A` and `BA=B` then `(A^(2010)+B^(2010))^(2011)` equals.

Options

A. `A+B`
B. `2010(A+B)`
C. `2011(A+B)`
D. `2^(2011)(A+B)`

Answer

D

Explanation

`(d)` Given `AB=A` and `BA=B`
`implies{:(A^(2)=A),(B^(2)=B):}`
`implies{:(A^(n)=A),(B^(n)=B):}`
`implies(A^(2010)+B^(2010))^(2011)=(A+B)^(2011)`
Now `(A+B)^(2)=A^(2)+B^(2)+AB+BA`
`=2(A+B)`
`implies(A+B)^(k)=2^(k)(A+B)`
`implies(A^(2010)+B^(2010))^(2011)=(A+B)^(2011)=2^(2011)(A+B)`