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In Exercises 31–36, mention an appropriate theorem in your explanation.Let $U$ be a square matrix such that $U^{T} U=I .$ Show that $\operatorname{det} U=\pm 1$

see the proof

Algebra

Chapter 3

Determinants

Section 2

Properties of Determinants

Introduction to Matrices

Baylor University

Idaho State University

Lectures

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In mathematics, the absolu…

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were given a statement and rest to prove this statement using the're, um from this section let you be a square matrix such that you transpose times you is equal to I were asked to show that the determinant of you is plus or minus one. Well, we have by fear, um six. And by the're, um five. It follows that the determinant of u transpose times you This is equal to the determinant of u transpose times the determinant of you the're, um six and then by the're, um five. This is equal to the determinants of you times the determinant of you. So the determinant of you squared now because you transpose u equals I. It follows that the determinant of u transpose you is also equal to the determinant of I which is one and therefore it follows that the determinant of you squared is equal toe one. Since the determinant of you is a real number, it follows that the determine interview is equal to plus or minus one

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