Question
(a) A matrix B in Mnxn(R) is called orthogonal if BB' = I. Prove that if B is
orthogonal, then det(B) = ±1.
(b) A matrix B in Mnx n(C) is called unitary if BB* = I, where (B*);} = B#,
the complex conjugate of B. Prove that if B is unitary, then
|det(B)| = 1. Hint: Prove that det(B)
%3D
det(B) where (B);j= Bj.
%3D
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Transcribed Image Text

(a) A matrix B in Mnxn(R) is called orthogonal if BB' = I. Prove that if B is orthogonal, then det(B) = ±1. (b) A matrix B in Mnx n(C) is called unitary if BB* = I, where (B*);} = B#, the complex conjugate of B. Prove that if B is unitary, then |det(B)| = 1. Hint: Prove that det(B) %3D det(B) where (B);j= Bj. %3D

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