If A and B are symmetric and commute, then which of the following is/are symmetric ?

Options

A. Both `A^(-1)B` and `A^(-1)B^(-1)` are symmetric.
B. `A^(-1)B` is symmetric but `A^(-1)B^(-1)` is not symmetric.
C. `A^(-1)B^(-1)` is symmetric but `A^(-1)B` is not symmetric.
D. Neither `A^(-1)B` nor `A^(-1)B^(-1)` are symmetric

Answer

A

Explanation

`(a)` `AB=BA`
Pre and post multiplying both sides by `A^(-1)`
`impliesA^(-1)(AB)A^(-1)=A^(-1)(BA)A^(-1)`
`implies(A^(-1)A)BA^(-1)=A^(-1)B(A A^(-1))`
`impliesBA^(-1)=A^(-1)B`
`implies(BA^(-1))^(T)=(A^(-1))^(T)B^(T)=A^(-1)B`( since `A` is symmetric , `:. A^(-1)` is also symmetric )
Thus `(A^(-1)B)^(T)=A^(-1)B`
`impliesA^(-1)B` is symmetric
`(A^(-1)(B^(-1))^(T)=((BA)^(-1))^(T)`
`=((AB)^(-1))^(T)`
`=((AB)^(T))^(-1)`
`=(B^(T)A^(T))^(-1)`
`=(BA)^(-1)`
`=A^(-1)B^(-1)`
`impliesA^(-1)B^(-1)` is also symmetric