If the matrix `A` and `B` are of `3xx3` and `(I-AB)` is invertible, then which of the following statement is/are correct ?

Question

If the matrix `A` and `B` are of `3xx3` and `(I-AB)` is invertible, then which of the following statement is/are correct ?

If the matrix `A` and `B` are of `3xx3` and `(I-AB)` is invertible, then which of the following statement is/are correct ?
A. `I-BA` is not invertible
B. `I-BA` is invertible
C. `I-BA` has for its inverse `I+B(I-AB)^(-1)A`
D. `I-BA` has for its inverse `I+A(I-BA)^(-1)B`

Monjur Alahi

5 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 - B::C
`(b,c)` Let `(I-AB)^(-1)=P`
`impliesP(I-AB)=I`
`impliesP-PAB=I`
`impliesPB^(-1)-PA=B^(-1)`
`impliesBPB^(-1)-BPA=I`
`impliesBPB^(-1)=I+BPA`
Now `BPB^(-1)=B(I-AB)^(-1)B^(-1)`
`=B(B(I-AB))^(-1)`
`=(B^(-1))^(-1)(B(I-AB))^(-1)`
`=(B(I-AB)B^(-1))^(-1)`
`=((B-BAB)B^(-1))^(-1)`
`=(I-BA)^(-1)`

5 views Answered 2023-02-05 15:29