Q16E Consider the matrices A For whic... [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: Q16E (page 380)

Consider the matrices A For which real numbers k is the zero state a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$
$A = [\begin{matrix} 0.1 k \\ 0.30.3 \end{matrix}]$

Short Answer

Expert verified

The Value of the matrix = $\frac{鈭� 97}{30} < k < \frac{21}{10}$

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 Value A

$\begin{matrix} \det (A 鈭� 位 I) = 0 \\ | \begin{matrix} 0.1 鈭� 位 & k \\ 0.3 & 0.3 鈭� 位 \end{matrix} | = 0 \\ (0.1 鈭� 位) (0.3 鈭� 位) 鈭� 0.3 k = 0 \\ 位^{2} 鈭� 0.4 位 + (0.03 鈭� 0.3 k) = 0 \\ 位_{1, 2} = \frac{0.4 卤 \sqrt{0.16 鈭� 4 (0.03 鈭� 0.3 k)}}{2} = 0.2 卤 \frac{\sqrt{0.04 + 1.2 k}}{2} \end{matrix}$

Solving the Equationk≥-130

$\begin{matrix} \frac{\sqrt{0.04 + 1.2 k}}{2} < 0.8 \\ \sqrt{0.04 + 1.2 k} < 1.6 \\ 0.04 + 1.2 k < 2.56 \\ k < \frac{21}{10} \end{matrix}$

Solving the Equation In case on the other handk<-130

$\begin{matrix} 0.04 + \frac{鈭� 0.04 鈭� 1.2 k}{4} < 1 \\ 0.12 鈭� 1.2 k < 4 \\ k > \frac{鈭� 97}{30} \end{matrix}$

The Equilibrium will be stable at $\frac{鈭� 97}{30} < k < \frac{21}{10}$

$\frac{鈭� 97}{30} < k < \frac{21}{10}$

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91影视

Consider the matrices A For which real numbers k is the zero state a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$
$A = [\begin{matrix} 0.1 k \\ 0.30.3 \end{matrix}]$

Short Answer

Step by step solution

Define Matrix

Calculation of the Value A

Solving the Equationk≥-130

Solving the Equation In case on the other handk<-130

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91影视

Consider the matrices A For which real numbers k is the zero state a stable equilibrium of the dynamical systemx鈫�(t+1)=Ax鈫�(t)A=[0.1k0.30.3]

Short Answer

Step by step solution

Define Matrix

Calculation of the Value A

Solving the Equationk≥-130

Solving the Equation In case on the other handk<-130

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Discrete Mathematics

Pure Maths

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider the matrices A For which real numbers k is the zero state a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$
$A = [\begin{matrix} 0.1 k \\ 0.30.3 \end{matrix}]$