If A is a nonsingular matrix such that `A A^(T)=A^(T)A` and `B=A^(-1) A^(T)`, then matrix B is

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

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

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

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`

4 views Answered 2023-02-05 15:29