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If the matrix A and B are of 3xx3 and (I...

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`

Text Solution

Verified by Experts

The correct Answer is:
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)`
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