Let A and B be two square matrices satisfying `A+BA^(T)=I` and `B+AB^(T)=I` and `O` is null matrix then identity the correct statement.

Options

A. `A=B^(T)`
B. `B=A^(T)`
C. `A^(4)-2A^(2)+A=O`
D. `A^(4)-2A^(2)-A=O`

Answer

A, B, C

Explanation

`A+BA^(T)=I` and `B+AB^(T)=I`
`impliesA^(T)+AB^(T)=I` and `B^(T)+BA^(T)=I`
`impliesA^(T)+I-B=I` and `B^(T)+I-A=I`
`impliesA^(T)=B` and `B^(T)=A`
Now `A+BA^(T)=I` and `B+AB^(T)=I`
`impliesA+B^(2)=I` and `B+A^(2)=I`
`impliesA+(I-A^(2))^(2)=I`