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

Options

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`

Answer

B, C

Explanation

`(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)`