Section 3.4
by C. Henry Edwards,
David E. Penney,
David T. Calvis,
Differential Equations and Linear Algebra, 4th Edition
- Differential Equations & Linear Algebra
- Contents
- Application Modules
- Preface
- 1 First-Order Differential Equations
- 2 Mathematical Models and Numerical Methods
- 2.1 Population Models
- 2.2 Equilibrium Solutions and Stability
- 2.3 Acceleration–Velocity Models Acceleration–Velocity Models
- 2.4 Numerical Approximation: Euler’s Method Numerical Approximation: Euler’s Method
- 2.5 A Closer Look at the Euler Method
- An Improvement in Euler’s Method
- 2.6 The Runge–Kutta Method The Runge–Kutta Method
- 3 Linear Systems and Matrices
- 4 Vector Spaces
- 5 Higher-Order Linear Differential Equations
- 5.1 Introduction: Second-Order Linear Equations
- 5.2 General Solutions of Linear Equations
- 5.3 Homogeneous Equations with Constant Coefficients
- 5.4 Mechanical Vibrations
- 5.5 Nonhomogeneous Equations and Undetermined Coefficients
- 5.6 Forced Oscillations and Resonance
- 6 Eigenvalues and Eigenvectors
- 7 Linear Systems of Differential Equations
- 7.1 First-Order Systems and Applications
- 7.2 Matrices and Linear Systems
- 7.3 The Eigenvalue Method for Linear Systems
- 7.4 A Gallery of Solution Curves of Linear Systems
- Systems of Dimension n=2
- Real Eigenvalues
- Saddle Points
- Nodes: Sinks and Sources
- Zero Eigenvalues and Straight-Line Solutions
- Repeated Eigenvalues; Proper and Improper Nodes
- The Special Case of a Repeated Zero Eigenvalue
- Complex Conjugate Eigenvalues and Eigenvectors
- Pure Imaginary Eigenvalues: Centers and Elliptical Orbits
- Complex Eigenvalues: Spiral Sinks and Sources
- A 3-Dimensional Example
- 7.4 Problems
- 7.4 Application Dynamic Phase Plane Graphics
- 7.5 Second-Order Systems and Mechanical Applications*
- 7.6 Multiple Eigenvalue Solutions
- 7.7 Numerical Methods for Systems
- 7.7 Application Comets and Spacecraft
- 8 Matrix Exponential Methods
- 9 Nonlinear Systems and Phenomena
- 9.1 Stability and the Phase Plane
- 9.2 Linear and Almost Linear Systems
- 9.3 Ecological Models: Predators and Competitors
- 9.4 Nonlinear Mechanical Systems
- 10 Laplace Transform Methods
- 11 Power Series Methods
- References for Further Study
- APPENDIX A Existence and Uniqueness of Solutions
- APPENDIX B Theory of Determinants
- Answers to Selected Problems
- Index
Section 3.4
1.
[518−155] 2.
[16−26−922−1815] 3.
⎡⎣⎢−26122220−618⎤⎦⎥ 4.
⎡⎣⎢445335−221021−20−2694⎤⎦⎥ 5.
AB=[−9−10112], BA=[−21185] 6.
AB=⎡⎣⎢72311−1310−8−244157⎤⎦⎥, BA=⎡⎣⎢11227−1716−21−22757⎤⎦⎥ 7.
AB=[26], BA=⎡⎣⎢345681091215⎤⎦⎥ 8.
AB=[2135150], BA=⎡⎣⎢37160−20−2591338⎤⎦⎥ 9.
BA=⎡⎣⎢47−22⎤⎦⎥ but AB is not defined.10.
AB=[15−2−61331] but BA is not defined.11.
AB=[11153] but BA is not defined.12. Neither product matrix AB nor BA is defined.
13.
A(BC)=(AB)C=[32−251−17] 14.
A(BC)=(AB)C=[−3] 15.
A(BC)=(AB)C=[1281510] 16.
A(BC)=(AB)C=⎡⎣⎢−4−9−14−4−12−18−2−9−1321217⎤⎦⎥ 17.
x=s(5, −2, 1, 0)+t(−4, 7, 0, 1) 18.
x=s(3, 1, 0, 0)+t(−6, 0, −9, 1) 19.
x=s(−3, 2, −1, 1, 0)+t(1, −6, 8, 0, 1) 20.
x=s(3, 1, 0, 0, 0)+t(−7, 0, 2, 10, 0) 21.
x=r(1, −2, 1, 0, 0)+s(−2, 3, 0, 1, 0 )+t(−7, −4, 2, 0, 1) 22.
x=r(1, 1, 0, 0, 0)+s(−7, 0, 1, 1, 0 )+t(−3, 0, 2, 0, 1) 23.
B=[2−3−12] 24.
B=[7−5−43] 25.
B=[3−2−75] 29.
A2=[a2+bcac+dcab+bdbc+d2], A2+(ad−bc)I=[a2+adac+dcab+bdd2+ad]=[a(a+d)c(a+d)b(a+d)d(a+d)]=(a+d)A 31. (a)
(A+B)(A−B)=[−25−71−34−34]≠[−8−44−45−51]=A2−B2 33.
[1001], [−100−1], [+100−1], [−100+1] 35.
[22−1−1] 37.
[0−110] 39.
A(c1x1+c2x2)=c1Ax1+c2Ax2=c1 ⋅0+c2 ⋅0=0 43.
A2=⎡⎣⎢6−3−3−36−3−3−36⎤⎦⎥=3A
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