If the square matrices A and B are such that `AB = A` and `BA = B`, then

Options

A. `(A^(T))^(2)=A^(T)`
B. `(A^(T))^(2)=B^(T)`
C. `(A^(T))^(2)=(A^(-1))^(-1)`
D. None of the above

Answer

A

Explanation

Let A and B be two matrices such that AB = A and BA = B
Now, consider AB = A
Take Transpose on both side
`(AB)^(T) = A^(T)`
`rArr A^(T) = B^(T).A^(T)" "...(1)`
Now, BA = B
Take, Transpose on both side
`(BA)^(T) = B^(T)`
`rArr B^(T) = A^(T).B^(T)" "...(2)`
Now, from equation (1) and (2). we have
`A^(T) = (A^(T).B^(T))A^(T)`
`A^(T)=A^(T)(B^(T)A^(T))`
`=A^(T)(AB)^(T)" "(because (AB)^(T) = B^(T)=B^(T)A^(T))`
`=A^(T).A^(T)`
Thus, `A^(T) = (A^(T))^(2)`