Let A and B be two matrices such that AB = A and BA = B. Which of the following statements are correct ?
1. `A^(2) = A`
2. `B^(2)=B`
3. `(AB)^(2)=AB`
Select the correct answer using the code given below :

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Answer

D

Explanation

We have, `AB=A`
`therefore" "A^(2)=(AB).(AB)=A.(BA)B`
`ABB" "(becauseBA=B)`
`=AB=A" "(because AB=A)`
Also, `B^(2)=(BA).(BA)=B.(AB).A`
`=B.A.A" "(because BA=B)`
Again, `(AB)^(2)=(AB).(AB)=A.(BA)B`
`-A.B.B" "(because BA=B)`
`A.B=A" "(because AB=A)`