If `A` and `B` are two non singular matrices and both are symmetric and commute each other, then

Options

A. `A^(-1) B`
B. `AB^(-1)`
C. `A^(-1) B^(-1)`
D. none of these

Explanation

Given that A and B commute, we have
`AB=BA" "( :' " A and B are symmetric")` (1)
Also,
`A^(T)=A, B^(T)=B` (2)
`(A^(-1) B)^(T)=B^(T)(A^(-1))^(T)=BA^(-1)`
( `:'` if A is symmetric, `A^(-1)` is also symmetric)
Also from Eq. (1),
`ABA^(-1)=B` (3)
or `A^(-1) ABA^(-1)=A^(-1)B`
or `IBA^(-1)=A^(-1)B`
or `BA^(-1)=A^(-1)B`
Hence, from Eq. (2),
`(A^(-1) B)^(T)=A^(-1) B`
Thus, `A^(-1)B` is symmetric. Similarly, `AB^(-1)` is also symmetric. Also,
`BA=AB`
or `(BA)^(-1)=(AB)^(-1)`
or `A^(-1)B^(-1)=B^(-1)A^(-1)`
or `(A^(-1) B^(-1))^(T)=(B^(-1)A^(-1))^(T)`
`=(A^(-1))^(T)(B^(-1))^(T)`
`=A^(-1)B^(-1)`
Hence, `A^(-1) B^(-1)` is symetric.