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1. For which values of a and b is the following system of equations inconsistent. x + 2y3z=4 3x - y + 5z = 2 4x + y + az = b (a) a= 2 and b = 6; (d) a = 1 and b = 3; (b) a = 2 and b‡6; (e) None of these. (c) a ‡ 2 and b = 6;

1. For which values of a and b is the following system of equations inconsistent. x + 2y3z=4 3x - y + 5z = 2 4x + y + az = b (a) a= 2 and b = 6; (d) a = 1 and b = 3; (b) a = 2 and b‡6; (e) None of these. (c) a ‡ 2 and b = 6;

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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1. 2. 3. For which values of a and b is the following system of equations inconsistent. x+2y3z = 4 3x = y + 5z = 2 4x + y + az = b (a) a= 2 and b = 6; (d) a = 1 and b = 3; (d) A = Find the standard matrix for the operator on R² which contracts with factor 1/4, then reflects about the line y = x. 0 (a) A = 1/4 0 (₁/11) ( 0 1/4 1/4 ¹/4) 0 (b) a = 2 and b = 6; (e) None of these. (c) a 2 and b = 6; 0 1/4 - (¹/4) 0 (e) None of these (b) A = (e) None of these (c) A = The linear operator T : R³ → R³ is defined by T(x₁, x2, X3) = (W₁, W2, W3), where w₁ = 2x₁ + 4x2 + x3; W₂ = 9x2+2x3; W3 = 2x1 8x2 - 2x3. Which of the following is correct. (a) T is not one to one. (b) T is one to one but the standard matrix for T-¹ does not exist. (c) T is one to one and its standard matrix for T-¹ is (d) T is one to one and its standard matrix for T-¹ is HOLI 0 1 (88) 0 3 0 1 3 3 WIN - WIN 3 0 1 3 1 -4 3 -3 2 -3 1623
Transcribed Image Text:1. 2. 3. For which values of a and b is the following system of equations inconsistent. x+2y3z = 4 3x = y + 5z = 2 4x + y + az = b (a) a= 2 and b = 6; (d) a = 1 and b = 3; (d) A = Find the standard matrix for the operator on R² which contracts with factor 1/4, then reflects about the line y = x. 0 (a) A = 1/4 0 (₁/11) ( 0 1/4 1/4 ¹/4) 0 (b) a = 2 and b = 6; (e) None of these. (c) a 2 and b = 6; 0 1/4 - (¹/4) 0 (e) None of these (b) A = (e) None of these (c) A = The linear operator T : R³ → R³ is defined by T(x₁, x2, X3) = (W₁, W2, W3), where w₁ = 2x₁ + 4x2 + x3; W₂ = 9x2+2x3; W3 = 2x1 8x2 - 2x3. Which of the following is correct. (a) T is not one to one. (b) T is one to one but the standard matrix for T-¹ does not exist. (c) T is one to one and its standard matrix for T-¹ is (d) T is one to one and its standard matrix for T-¹ is HOLI 0 1 (88) 0 3 0 1 3 3 WIN - WIN 3 0 1 3 1 -4 3 -3 2 -3 1623
4. Let c = (1,1,1,1), c = (2, 2+x, 2, 2), c = (3, 3,3 + x, 3) and c (3, 4, 4, 4+ x). Find all x € (-∞, ∞) such that {c1, c₂, c²,c} is linearly dependent. E (a) x = 1 (b) x = 0 (c) x = -1 (d) x = 0 or x = −1 (e) None of these 5. 7. 8. Let = 2 be an eigenvalue of an n x n matrix A. Which of the following is correct? (a) 21 · A| ‡ 0 (e) None of these 6. Assume that a 3 × 3 matrix A has three eigenvalues A₁, A2 and X3. 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 (b) 21 – A is invertible; (c) The rank of 21 – A is strictly less than n; (d) The eigenvector of A corresponding to λ = 2 is zero. The set of all the solutions of the equation z³ = = is (a) {-3} (b) {-3,1-√3i, 1+√3i} (c) {-3, — B³i, } + ³ i} {−3} 1 2 (d) {-3, 2-³√³i, 3 + ³√³₁} (e) None of these -27 in the complex plane Express 1+√3i in polar form using its principal argument. (b) 2 (cos (π/6) +isin(π/6)) (d) 2(cos(-π/6) +isin(-π/6)) (a) 2(cos(-π/3) + i sin(-π/3)) (c) 2(cos(π/3) + i sin(π/3)) (e) None of these
Transcribed Image Text:4. Let c = (1,1,1,1), c = (2, 2+x, 2, 2), c = (3, 3,3 + x, 3) and c (3, 4, 4, 4+ x). Find all x € (-∞, ∞) such that {c1, c₂, c²,c} is linearly dependent. E (a) x = 1 (b) x = 0 (c) x = -1 (d) x = 0 or x = −1 (e) None of these 5. 7. 8. Let = 2 be an eigenvalue of an n x n matrix A. Which of the following is correct? (a) 21 · A| ‡ 0 (e) None of these 6. Assume that a 3 × 3 matrix A has three eigenvalues A₁, A2 and X3. 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 (b) 21 – A is invertible; (c) The rank of 21 – A is strictly less than n; (d) The eigenvector of A corresponding to λ = 2 is zero. The set of all the solutions of the equation z³ = = is (a) {-3} (b) {-3,1-√3i, 1+√3i} (c) {-3, — B³i, } + ³ i} {−3} 1 2 (d) {-3, 2-³√³i, 3 + ³√³₁} (e) None of these -27 in the complex plane Express 1+√3i in polar form using its principal argument. (b) 2 (cos (π/6) +isin(π/6)) (d) 2(cos(-π/6) +isin(-π/6)) (a) 2(cos(-π/3) + i sin(-π/3)) (c) 2(cos(π/3) + i sin(π/3)) (e) None of these
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