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. (a) If A and B are matrices and AB = 0 (the zero matrix), then A=0 or B = 0. SOMETIMES ALWAYS (b) If X(t) is a solution to the linear system X' = AX, then eX(t) is also a solution. NEVER ALWAYS (c) If a 2 x 2 matrix A has real distinct eigenvalues and det A = 1, then 0 < trace(A) <2. ALWAYS ALWAYS SOMETIMES (e) If M = [2 SOMETIMES (d) If det A = 0, then the system X'= AX has more than two equilibrium solutions. ALWAYS NEVER SOMETIMES NEVER SOMETIMES 2] 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. NEVER 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. (a) If A and B are matrices and AB = 0 (the zero matrix), then A=0 or B = 0. SOMETIMES ALWAYS (b) If X(t) is a solution to the linear system X' = AX, then eX(t) is also a solution. NEVER ALWAYS (c) If a 2 x 2 matrix A has real distinct eigenvalues and det A = 1, then 0 < trace(A) <2. ALWAYS ALWAYS SOMETIMES (e) If M = [2 SOMETIMES (d) If det A = 0, then the system X'= AX has more than two equilibrium solutions. ALWAYS NEVER SOMETIMES NEVER SOMETIMES 2] 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. NEVER 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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Transcribed Image Text: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.
(a) If A and B are matrices and AB = 0 (the zero matrix), then A = 0 or B = 0.
SOMETIMES
ALWAYS
(b) If X(t) is a solution to the linear system X'= AX, then et X(t) is also a solution.
ALWAYS
ALWAYS
SOMETIMES
(c) If a 2 x 2 matrix A has real distinct eigenvalues and det A = 1, then 0 < trace(A) < 2.
ALWAYS
=
SOMETIMES
ALWAYS
NEVER
(d) If det A = 0, then the system X' = AX has more than two equilibrium solutions.
SOMETIMES
NEVER
NEVER
(e) If M
MX(t) is also a solution to the system X'= AX.
SOMETIMES
NEVER
and X(t) is a solution to the planar linear system X' = AX, then Y(t) =
NEVER
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