6. Let S = {(a, b)lab >0}. Sketch the set S and determine if it is a subspace of R². 7. If A is a 3 x 3 matrix with |A| = -2, compute the following determinnants: |(49)³| |2A'(A-¹)²| 8. Determine the values of for which the following system of equations has nontrivial solutions. Find the solutions for each value of 2. 0 0 (5-2)x₁ 4x1 2x1 (a) u + v (c) u - 2w (e) 2u - 5v - w + 4x2 + (5-2)x2 + 2x2 + (2-2)x3 9. Compute the following for u = (3,-1, 5), v = (2, 3, 7), w = (0, 1, -3) or state it is not a valid expression. 2x3 = 0 2x3 = 0 = 0 (b) 3u + v (d) 4u2v + 3w (f) u. v +2w
6. Let S = {(a, b)lab >0}. Sketch the set S and determine if it is a subspace of R². 7. If A is a 3 x 3 matrix with |A| = -2, compute the following determinnants: |(49)³| |2A'(A-¹)²| 8. Determine the values of for which the following system of equations has nontrivial solutions. Find the solutions for each value of 2. 0 0 (5-2)x₁ 4x1 2x1 (a) u + v (c) u - 2w (e) 2u - 5v - w + 4x2 + (5-2)x2 + 2x2 + (2-2)x3 9. Compute the following for u = (3,-1, 5), v = (2, 3, 7), w = (0, 1, -3) or state it is not a valid expression. 2x3 = 0 2x3 = 0 = 0 (b) 3u + v (d) 4u2v + 3w (f) u. v +2w
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematics Practice Problems**
**6. Problem Statement**
Let \( S = \{(a, b) | ab > 0\} \). Sketch the set \( S \) and determine if it is a subspace of \( \mathbb{R}^2 \).
**7. Determinant Computation**
If \( A \) is a 3 x 3 matrix with \( |A| = -2 \), compute the following determinants:
- \( |(A^3)^3| \)
- \( |2A^4(A^{-1})^2| \)
**8. System of Equations**
Determine the values of \( \lambda \) for which the following system of equations has nontrivial solutions. Find the solutions for each value of \( \lambda \).
\[
\begin{align*}
(5 - \lambda)x_1 + 4x_2 + 2x_3 &= 0 \\
4x_1 + (5 - \lambda)x_2 + 2x_3 &= 0 \\
2x_1 + 2x_2 + (2 - \lambda)x_3 &= 0 \\
\end{align*}
\]
**9. Vector Computation**
Compute the following for \( \mathbf{u} = (3, -1, 5) \), \( \mathbf{v} = (2, 3, 7) \), \( \mathbf{w} = (0, 1, -3) \) or state it is not a valid expression.
(a) \( \mathbf{u} + \mathbf{v} \)
(b) \( 3\mathbf{u} + \mathbf{v} \)
(c) \( \mathbf{u} - 2\mathbf{w} \)
(d) \( 4\mathbf{u} - 2\mathbf{v} + 3\mathbf{w} \)
(e) \( 2\mathbf{u} - 5\mathbf{v} - \mathbf{w} \)
(f) \( \mathbf{u} \cdot \mathbf{v} + 2\mathbf{w} \)
(g) \( 2\mathbf{u} - 3\mathbf{v} + \|\mathbf{v}\|\|\mathbf{w}\| \)
(h) \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05eef005-fd90-429d-9eeb-a5961680e45b%2Fcd356b1e-241f-4b00-9402-f3c9193ea290%2Fxzrvjy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Mathematics Practice Problems**
**6. Problem Statement**
Let \( S = \{(a, b) | ab > 0\} \). Sketch the set \( S \) and determine if it is a subspace of \( \mathbb{R}^2 \).
**7. Determinant Computation**
If \( A \) is a 3 x 3 matrix with \( |A| = -2 \), compute the following determinants:
- \( |(A^3)^3| \)
- \( |2A^4(A^{-1})^2| \)
**8. System of Equations**
Determine the values of \( \lambda \) for which the following system of equations has nontrivial solutions. Find the solutions for each value of \( \lambda \).
\[
\begin{align*}
(5 - \lambda)x_1 + 4x_2 + 2x_3 &= 0 \\
4x_1 + (5 - \lambda)x_2 + 2x_3 &= 0 \\
2x_1 + 2x_2 + (2 - \lambda)x_3 &= 0 \\
\end{align*}
\]
**9. Vector Computation**
Compute the following for \( \mathbf{u} = (3, -1, 5) \), \( \mathbf{v} = (2, 3, 7) \), \( \mathbf{w} = (0, 1, -3) \) or state it is not a valid expression.
(a) \( \mathbf{u} + \mathbf{v} \)
(b) \( 3\mathbf{u} + \mathbf{v} \)
(c) \( \mathbf{u} - 2\mathbf{w} \)
(d) \( 4\mathbf{u} - 2\mathbf{v} + 3\mathbf{w} \)
(e) \( 2\mathbf{u} - 5\mathbf{v} - \mathbf{w} \)
(f) \( \mathbf{u} \cdot \mathbf{v} + 2\mathbf{w} \)
(g) \( 2\mathbf{u} - 3\mathbf{v} + \|\mathbf{v}\|\|\mathbf{w}\| \)
(h) \(
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