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5. 3 C d S a b e porld at t Let λ = 2 be an eigenvalue of an n is correct? (a) |21 - A| ‡0 (b) 21 - A is invertible; (c) The (e) None of these (d) The eigenvector of A corresp a b C d e

5. 3 C d S a b e porld at t Let λ = 2 be an eigenvalue of an n is correct? (a) |21 - A| ‡0 (b) 21 - A is invertible; (c) The (e) None of these (d) The eigenvector of A corresp a b C d e

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter7: Triangles
Section: Chapter Questions
Problem 1RP: We mentioned in Section 7.5 that our algebraic treatment of vectors could be attributed, in part, to...
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**Educational Website: Math Problems** ### Problem 4 Given the vectors: \[ \overrightarrow{v_1} = (1,1,1),\, \overrightarrow{v_2} = (2, 2 + x, 2x),\, \overrightarrow{v_3} = (3, 3, 3)\] and: \[ \overrightarrow{v_4} = (3, 4 + x, 4) \] Find all \( x \in (-\infty, \infty) \) such that \(\{ \overrightarrow{v_1}, \overrightarrow{v_2}, \overrightarrow{v_3}, \overrightarrow{v_4} \}\) is linearly dependent. Options: - \( (a) \ x = 1 \) - \( (b) \ x = 0 \) - \( (c) \ x = -1 \) - \( (d) \ x = 0 \text{ or } x = -1 \) - \( (e) \ None of these \) ### Problem 5 Let \( \lambda = 2 \) be an eigenvalue of an \( n \times n \) matrix \( A \). Which of the following is correct? Options: - \( (a) |2I - A| \neq 0 \) - \( (b) 2I - A \text{ is invertible} \) - \( (c) The rank of \( 2I - A \) is strictly less than \( n \) - \( (d) The eigenvector of \( A \) corresponding to \( \lambda = 2 \) is zero - \( (e) None of these **Note:** In this context, "I" denotes the identity matrix of the same dimension as \(A\).
Transcribed Image Text:**Educational Website: Math Problems** ### Problem 4 Given the vectors: \[ \overrightarrow{v_1} = (1,1,1),\, \overrightarrow{v_2} = (2, 2 + x, 2x),\, \overrightarrow{v_3} = (3, 3, 3)\] and: \[ \overrightarrow{v_4} = (3, 4 + x, 4) \] Find all \( x \in (-\infty, \infty) \) such that \(\{ \overrightarrow{v_1}, \overrightarrow{v_2}, \overrightarrow{v_3}, \overrightarrow{v_4} \}\) is linearly dependent. Options: - \( (a) \ x = 1 \) - \( (b) \ x = 0 \) - \( (c) \ x = -1 \) - \( (d) \ x = 0 \text{ or } x = -1 \) - \( (e) \ None of these \) ### Problem 5 Let \( \lambda = 2 \) be an eigenvalue of an \( n \times n \) matrix \( A \). Which of the following is correct? Options: - \( (a) |2I - A| \neq 0 \) - \( (b) 2I - A \text{ is invertible} \) - \( (c) The rank of \( 2I - A \) is strictly less than \( n \) - \( (d) The eigenvector of \( A \) corresponding to \( \lambda = 2 \) is zero - \( (e) None of these **Note:** In this context, "I" denotes the identity matrix of the same dimension as \(A\).
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