If A is an `3xx3` non-singular matrix such that `A A^T=A^TA and B=A^(-1)A^T," then " B B^T` equals

Options

A. `B^(-1)`
B. `(B^(-1))^T`
C. `I+B`
D. `{:[(2015,0),(1,2015)]:}`

Answer

D

Explanation

We have,
`:. B B^T=(A^(-1)A^T)(A^(-1)A^T)^T`
`=(A^(-1)A^T)((A^T)^T(A^(-1))^T)`
`=(A^(-1)A^T)(A(A^(-1))^T) [:'(A^T)^T=A]`
`=A^(-1)(A^TA)(A^(-1))^T`
`=A^(-1)(A A^T)(A^T)^(-1) [:' A^T A=A A^T("given")and(A^(-1))^T=(A^T)^(-1)]`
`=(A^(-1)A)(A^T(A^T)^(-1))=I=I`