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Chapter 1 and 2

# Chapter 1 and 2 - Chapter 1 Introduction and Review 1 If A...

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Chapter 1 Introduction and Review 1. If A IR n × n and α is a scalar, what is det( α A )? What is det( - A )? Answer 1.1 Suppose A IR n × n and α IR . Denote the identity of IR n × n by I n . Then det( α A ) = det( α ( I n A )) = det(( α I n ) A ) = det( α I n ) det A = α n det A . Thus det( - A ) = det(( - 1) A ) = ( - 1) n det A . 2. If A is orthogonal, what is det A ? If A is unitary, what is det A ? Answer 1.2 A is orthogonal = AA T = I = 1 = det I = det( AA T ) = det A det A T = det A det A = (det A ) 2 = det A = ± 1 . A is unitary = AA H = I = 1 = det I = det( AA H ) = det A det A = ( re j θ )( re - j θ ) = r 2 for some θ , r IR = r = ± 1 , and so det A = e j θ for some θ IR . 3. Let x, y IR n . Show that det( I - xy T ) = 1 - y T x . Answer 1.3 Using determinant properties 16 and 17, det( I n - xy T ) = det(1) det( I n - xy T ) = det I n x y T 1 = det( I n ) det(1 - y T x ) = 1 - y T x . 1

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2 CHAPTER 1. INTRODUCTION AND REVIEW 4. Let U 1 , U 2 , . . . , U k IR n × n be orthogonal matrices. Show that the product U = U 1 U 2 · · · U k is an orthogonal matrix.
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