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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

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Question

For the following systems, the origin is the equilibrium point.
a) Write each system in matrix form Ax.
b)
Determine the eigenvalues of A.
c) State whether the origin is a stable or unstable equilibrium.
d)
State whether the origin is a node, saddle point, spiral point, or center.
e) State the equations of the straight-line trajectories and tell whether they are going
towards or away from the origin. If none exist, state so.
f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to
solve the system. (See examples in Section 7.4)
1.
dx
dt
dy
dt
= -2x + 5cy
4y
=
dx
dt
C = 3

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For the following systems, the origin is the equilibrium point. a) Write each system in matrix form b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) 2. dx dt dy dt = 5x + 2y = 2x + 5y (Ctrl) dx dt = Ax.
For the following systems, the origin is the equilibrium point. a) Write each system in matrix form b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. 4. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) dx dt dy dt = x + 4y dt = 4x + y = Ax.
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