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6. Assume that a 3 x 3 matrix A has three eigenvalues A₁, A2 and A3. If A₁ = A₂ = 2, A3 = 6 and the trace of A is 5+x. Then x is equal to (a) 5 (b) 4; (c) 3; (d) 2; (e) None of these a b С d e The set of all the solutions of the equation z³ = -27 in the complex plane is (a) (-3) (b) {-3,1-√3i,1+√3i) (c) {-3, 1-i, + i) (d) {−3, 2 – ³√³i, ³2 + ³√3₁} (e) None of these - 7.

6. Assume that a 3 x 3 matrix A has three eigenvalues A₁, A2 and A3. If A₁ = A₂ = 2, A3 = 6 and the trace of A is 5+x. Then x is equal to (a) 5 (b) 4; (c) 3; (d) 2; (e) None of these a b С d e The set of all the solutions of the equation z³ = -27 in the complex plane is (a) (-3) (b) {-3,1-√3i,1+√3i) (c) {-3, 1-i, + i) (d) {−3, 2 – ³√³i, ³2 + ³√3₁} (e) None of these - 7.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Questions on Eigenvalues and Complex Numbers #### 6. Eigenvalues and Trace of a Matrix **Problem Statement:** Assume that a \(3 \times 3\) matrix \(A\) has three eigenvalues \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\). If \(\lambda_1 = \lambda_2 = 2\), \(\lambda_3 = 6\) and the trace of \(A\) is \(5 + x\), then \(x\) is equal to: - (a) 5 - (b) 4 - (c) 3 - (d) 2 - (e) None of these **Answer Options:** ``` a b c d e ``` #### 7. Solutions of a Complex Equation **Problem Statement:** The set of all the solutions of the equation \(z^3 = -27\) in the complex plane is: - (a) \(\{-3\}\) - (b) \(\{-3, 1 - \sqrt{3}i, 1 + \sqrt{3}i\}\) - (c) \(\{-3, \frac{1}{2} - \frac{\sqrt{3}}{2}i, \frac{1}{2} + \frac{\sqrt{3}}{2}i\}\) - (d) \(\{-3, \frac{3}{2} - \frac{3\sqrt{3}}{2}i, \frac{3}{2} + \frac{3\sqrt{3}}{2}i\}\) - (e) None of these
Transcribed Image Text:### Questions on Eigenvalues and Complex Numbers #### 6. Eigenvalues and Trace of a Matrix **Problem Statement:** Assume that a \(3 \times 3\) matrix \(A\) has three eigenvalues \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\). If \(\lambda_1 = \lambda_2 = 2\), \(\lambda_3 = 6\) and the trace of \(A\) is \(5 + x\), then \(x\) is equal to: - (a) 5 - (b) 4 - (c) 3 - (d) 2 - (e) None of these **Answer Options:** ``` a b c d e ``` #### 7. Solutions of a Complex Equation **Problem Statement:** The set of all the solutions of the equation \(z^3 = -27\) in the complex plane is: - (a) \(\{-3\}\) - (b) \(\{-3, 1 - \sqrt{3}i, 1 + \sqrt{3}i\}\) - (c) \(\{-3, \frac{1}{2} - \frac{\sqrt{3}}{2}i, \frac{1}{2} + \frac{\sqrt{3}}{2}i\}\) - (d) \(\{-3, \frac{3}{2} - \frac{3\sqrt{3}}{2}i, \frac{3}{2} + \frac{3\sqrt{3}}{2}i\}\) - (e) None of these
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