Q17E For the Matrices A find real clo... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q17E (page 380)

For the Matrices A find real closed formulas for the trajectory $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ where $\overset{鈫�}{x} (0) = [\begin{array}{l} 0 \\ 1 \end{array}]$
$A = [\begin{matrix} 0.6 - 0.8 \\ 0.80.6 \end{matrix}]$

Short Answer

Expert verified

The Value of the Matrix = $A^{t} x_{0} = [\begin{matrix} 鈭� \sin (t \arctan \frac{4}{3}) \\ \cos (t \arctan \frac{4}{3}) \end{matrix}]$

Step by step solution

Define Matrix

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

Calculation of the sum

$\begin{matrix} \det (A 鈭� 位 I) = 0 \\ | \begin{matrix} 0.6 鈭� 位 & 鈭� 0.8 \\ 0.8 & 0.6 鈭� 位 \end{matrix} | = 0 \\ {(0.6 鈭� 位)}^{2} + 0.64 = 0 \\ 位^{2} 鈭� 1.2 位 + 1 = 0 \end{matrix}$

$\begin{matrix} 位_{1, 2} = \frac{1.2 卤 \sqrt{1.44 鈭� 4}}{2} = 0.6 卤 0.8 i = \cos (\arctan (\frac{4}{3})) 卤 i \sin (\arctan (\frac{4}{3})) \\ E_{位_{1}} = s p a n ([\begin{matrix} 1 \\ 鈭� i \end{matrix}]) \\ 鈬� S = [\begin{matrix} 0 & 1 \\ 鈭� 1 & 0 \end{matrix}] \end{matrix}$

Solving the Equation to find the value

$A^{t} x_{0} = S [\begin{matrix} \cos (t \arctan \frac{4}{3}) & 鈭� \sin (t \arctan \frac{4}{3}) \\ \sin (t \arctan \frac{4}{3}) & \cos (t \arctan \frac{4}{3}) \end{matrix}] S^{鈭� 1} x_{0} = [\begin{matrix} 鈭� \sin (t \arctan \frac{4}{3}) \\ \cos (t \arctan \frac{4}{3}) \end{matrix}] .$

A^{t} x_{0} = [\begin{matrix} 鈭� \sin (t \arctan \frac{4}{3}) \\ \cos (t \arctan \frac{4}{3}) \end{matrix}]

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91影视

For the Matrices A find real closed formulas for the trajectory $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ where $\overset{鈫�}{x} (0) = [\begin{array}{l} 0 \\ 1 \end{array}]$
$A = [\begin{matrix} 0.6 - 0.8 \\ 0.80.6 \end{matrix}]$

Short Answer

Step by step solution

Define Matrix

Calculation of the sum

Solving the Equation to find the value

One App. One Place for Learning.

Most popular questions from this chapter

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Probability and Statistics

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91影视

For the Matrices A find real closed formulas for the trajectoryx鈫�(t+1)=Ax鈫�(t)where x鈫�(0)=[01]A=[0.6-0.80.80.6]

Short Answer

Step by step solution

Define Matrix

Calculation of the sum

Solving the Equation to find the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

For the Matrices A find real closed formulas for the trajectory $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ where $\overset{鈫�}{x} (0) = [\begin{array}{l} 0 \\ 1 \end{array}]$
$A = [\begin{matrix} 0.6 - 0.8 \\ 0.80.6 \end{matrix}]$