If `A ` and `B` are two non-singular matrices which commute, then `(A(A+B)^(-1)B)^(-1)(AB)=`

Question

If `A ` and `B` are two non-singular matrices which commute, then `(A(A+B)^(-1)B)^(-1)(AB)=`

If `A ` and `B` are two non-singular matrices which commute, then `(A(A+B)^(-1)B)^(-1)(AB)=`
A. `A+B`
B. `A^(-1)+B^(-1)`
C. `A^(-1)+B`
D. none of these

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

Salim Ahmed Kawsar

Correct Answer - A
`(a)` `(A(A+B)^(-1)B)^(-1)(AB)`
`=B^(-1)(A+B^(-1))^(-1)A^(-1)(AB)`
`=B^(-1)(A+B)A^(-1)(AB)`
`=(B^(-1)A+I)A^(-1)(AB)`
`=(B^(-1)A A^(-1)+A^(-1))(AB)`
`=(B^(-1)+A^(-1))(AB)`
`=B^(-1)AB+A^(-1)AB`
`=B^(-1)BA+A^(-1)AB`
`=A+B`

4 views Answered 2023-02-05 15:29