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If A is an 3xx3 non-singular matrix such...

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

A

`B^(-1)`

B

`(B^(-1))`

C

`I+B`

D

`I`

Text Solution

Verified by Experts

The correct Answer is:
D
`B = A^(-1) A'`
`B' = (A ^(-1) A')' = A(A^(-1))'`
Now, `BB' = (A^(-1)A') A(A^(-1)) ' = A^(-1) (A'A) (A^(-1))'`
`= A^(-1) (A A')(A^(-1)) " " [because A' A = A A' ]`
`= (A^(-1) A) A'(A^(-1))'`
`= (IA') (A^(-1))'=A^'(A^(-1))' = (A^(-1)A) ' = I' = I`
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