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d. Prove that |yz| = |y||z| for any y, z = C.

d. Prove that |yz| = |y||z| for any y, z = C.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 40E
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Problem 10 The conjugate of a complex number z = a + bi is a - bi. We denote the conjugate by z. We also define the modulus of a complex number z, denoted by [z], as √zz. Note that zz is always a nonnegative real number, so this is also a nonnegative real number. Let A be a complex n × n matrix, where n ≥ 2. We define the matrix A to be the matrix obtained by taking the conjugate of every entry of A. a. Calculate A, where A = = e. Prove that det A 4 - +21]. b. Calculate [2 – 4i|. c. Prove that for any complex numbers x and y, (xy) = (x)(y) and x + y = x+y. d. Prove that |yz| = |y||z| for any y, z € C. i 3+2i det A. (Hint: Show the 2x2 case and use induction on n.) f. A unitary matrix A € Mnxn(C) is one where (AT) = A−¹. Prove that if A is a unitary matrix, then | det A| = 1.
Transcribed Image Text:Problem 10 The conjugate of a complex number z = a + bi is a - bi. We denote the conjugate by z. We also define the modulus of a complex number z, denoted by [z], as √zz. Note that zz is always a nonnegative real number, so this is also a nonnegative real number. Let A be a complex n × n matrix, where n ≥ 2. We define the matrix A to be the matrix obtained by taking the conjugate of every entry of A. a. Calculate A, where A = = e. Prove that det A 4 - +21]. b. Calculate [2 – 4i|. c. Prove that for any complex numbers x and y, (xy) = (x)(y) and x + y = x+y. d. Prove that |yz| = |y||z| for any y, z € C. i 3+2i det A. (Hint: Show the 2x2 case and use induction on n.) f. A unitary matrix A € Mnxn(C) is one where (AT) = A−¹. Prove that if A is a unitary matrix, then | det A| = 1.
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