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

Question

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

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

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A
`(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