1. For each of the following statements, determine if the conclusion ALWAYS follows from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the conclusion is NEVER true given the assumptions. You do not need to show any work or justify your answers to these questions – only your circled answer will be graded.(a) If \( A \) and \( B \) are matrices and \( AB = 0 \) (the zero matrix), then \( A = 0 \) or \( B = 0 \).- ALWAYS- SOMETIMES- NEVER(b) If \( X(t) \) is a solution to the linear system \( X' = AX \), then \( e^t X(t) \) is also a solution.- ALWAYS- SOMETIMES- NEVER(c) If a \( 2 \times 2 \) matrix \( A \) has real distinct eigenvalues and \( \det A = 1 \), then \( 0 < \text{trace}(A) < 2 \).- ALWAYS- SOMETIMES- NEVER(d) If \( \det A = 0 \), then the system \( X' = AX \) has more than two equilibrium solutions.- ALWAYS- SOMETIMES- NEVER(e) If \( M = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \) and \( X(t) \) is a solution to the planar linear system \( X' = AX \), then \( Y(t) = MX(t) \) is also a solution to the system \( X' = AX \).- ALWAYS- SOMETIMES- NEVER

“Since you have posted a question with multiple sub parts, we will provide the solution only to the…

1. For each of the following statements, determine if the conclusion ALWAYS follows from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the conclusion is NEVER true given the assumptions. You do not need to show any work or justify your answers to these questions - only your circled answer will be graded. (a) If A and B are matrices and AB = 0 (the zero matrix), then A=0 or B = 0. ALWAYS SOMETIMES (b) If X(t) is a solution to the linear system X'= AX, then et X(t) is also a solution. ALWAYS SOMETIMES NEVER NEVER (c) If a 2 x 2 matrix A has real distinct eigenvalues and det A = 1, then 0 < trace(A) < 2. ALWAYS SOMETIMES NEVER

1. For each of the following statements, determine if the conclusion ALWAYS follows from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the conclusion is NEVER true given the assumptions. You do not need to show any work or justify your answers to these questions - only your circled answer will be graded. (a) If A and B are matrices and AB = 0 (the zero matrix), then A=0 or B = 0. ALWAYS SOMETIMES (b) If X(t) is a solution to the linear system X'= AX, then et X(t) is also a solution. ALWAYS SOMETIMES NEVER NEVER (c) If a 2 x 2 matrix A has real distinct eigenvalues and det A = 1, then 0 < trace(A) < 2. ALWAYS SOMETIMES NEVER

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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Question

1. For each of the following statements, determine if the conclusion ALWAYS follows from the assumptions, if the conclusion is SOMETIMES true given the assumptions, or if the conclusion is NEVER true given the assumptions. You do not need to show any work or justify your answers to these questions – only your circled answer will be graded.

(a) If \( A \) and \( B \) are matrices and \( AB = 0 \) (the zero matrix), then \( A = 0 \) or \( B = 0 \).

- ALWAYS
- SOMETIMES
- NEVER

(b) If \( X(t) \) is a solution to the linear system \( X' = AX \), then \( e^t X(t) \) is also a solution.

- ALWAYS
- SOMETIMES
- NEVER

(c) If a \( 2 \times 2 \) matrix \( A \) has real distinct eigenvalues and \( \det A = 1 \), then \( 0 < \text{trace}(A) < 2 \).

- ALWAYS
- SOMETIMES
- NEVER

(d) If \( \det A = 0 \), then the system \( X' = AX \) has more than two equilibrium solutions.

- ALWAYS
- SOMETIMES
- NEVER

(e) If \( M = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \) and \( X(t) \) is a solution to the planar linear system \( X' = AX \), then \( Y(t) = MX(t) \) is also a solution to the system \( X' = AX \).

- ALWAYS
- SOMETIMES
- NEVER

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