Exam 4 Review Guide Blank
Exam 4 Review Guide Blank
Exam 4 Review Guide Blank
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1. 1. If A = . a) How do we know that A–1 exists?
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b) Write Ax as a system of equations.
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c) Find A–1 and use it to solve Ax .
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2 x1 8 x4 6
2. a) Set up an augmented matrix for the system 2 x1 x2 4 x4 2 and completely reduce the
x3 5 x4 2
augmented matrix using exactly two row operations.
7. The equation that defines r as an eigenvalue of the matrix A with corresponding eigenvector ξ is
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8. Find the general solution of the system Dx x . Express the general solution both as a linear
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combination of solution vectors and in terms of (t).
10. Tanks A and B are interconnected. Tank A initially contains 15 lbs. of salt dissolved in 100 gallons
of water. Tank B initially contains 100 gallons of pure water. Pure water flows into tank A at the
rate of 3 gal/min. Brine flows from tank A to tank B at a rate of 4 gal/min. Brine flows from tank B
to tank A at a rate of 1 gal/min. Brine leaves the system from tank B at a rate of 5 gal/min. Let x1(t)
and x2(t) denote the amounts of salt in tanks A and B, respectively, at any time t. Set up a system of
equations and initial vector to model this system.
13. Use Laplace transforms to find the zero-state output of the mass-damper system modeled by
u ' ' 2u ' u t 3 e t .
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b) L 3
s 2s 2 s
s
c) L 1 2
s 4s 5