Question

If A is a nonsingular matrix such that `A A^(T)=A^(T)A` and `B=A^(-1) A^(T)`, then matrix B is
A. involuntary
B. orthogonal
C. idempotent
D. none of these

Answer 1

Correct Answer - B
Given,
`B=A^(-1) A^(T)`
`implies B^(T)=(A^(-1)A^(T))^(T)=Axx(A^(-1))^(T)`
`implies BxxB^(T)=A^(-1) A^(T)xxAxx(A^(-1))^(T)`
`=A^(-1)xx(A^(T)xxA)(A^(-1))^(T)`
`=A^(-1) (AxxA^(T)) (A^(-1))^(T)`
`=(A^(-1) A)xx(A^(-1) A)^(T)=IxxI^(T)=I`