(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

(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

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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