If A and B arę square matrices of same order such that AB = A and BA = B, then

Options

A. `B^2=B and A^2=A`
B. `B^2 ne B and A^2= A`
C. `A^2 ne A,B^2 =B`
D. `A^2 ne A, B^2 ne B`

Answer

A

Explanation

We have,
`A^2=A A`
`rArr A^2=(AB)A [:' AB=A]`
`rArrA^2=A(BA)`
`rArr A^2=AB [:' BA=B]`
`rArr A^2=A [:'AB=A]`
and,
`A^2=A A`
`rArr B^2=(BA)A [:' BA=B]`
`rArrB^2=A(AB)`
`rArr B^2=BA [:' AB=A]`
`rArr B^2=B [:'BA=B]`