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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q15E (page 355)

For the matrices A and the vectors $\overset{炉}{x_{0}}$ in Exercise 13 through 19, find closed formulas for $A^{t} 鈥� \overset{炉}{x_{0}}$ ,where t is an arbitrary positive integer. Follow the strategy outlined in Theorem $7.1.6$ and illustrated in Example 1. In Exercises 16 through 19,feel free to use technology.
15.role="math" localid="1668573864526" $A = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}], \overset{炉}{x_{0}} = [\begin{matrix} 1 \\ 0 \end{matrix}]$

Short Answer

Expert verified

The closed formula is,

A^{t} x_{0} = \frac{1}{12} [\begin{matrix} 2 + 4^{- t} \\ 4 - 4^{- t} \end{matrix}]

Step by step solution

01

Definition of matrices

A function is defined as a relationship between a set of inputs that each have one output.

Given,

$A = [0.5 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 0.25 0.5 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 0.75], \overset{炉}{x_{0}} = [1 0]$

The matrix A is the same as in Exercise 7.

So,we know that

$A^{t} = \frac{1}{12} [\begin{matrix} 2 + 4^{t} & 4^{t} - 1 \\ 4 - 4^{t} & - 4^{t} - 2 \end{matrix}]$

$A^{t} x_{0} = 鈥� \frac{1}{12} [\begin{matrix} 2 + 4^{t} & 4^{t} - 1 \\ 4 - 4^{t} & - 4^{t} - 2 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}]$

02

Multiply the matrices

Then,

A^{t} x_{0} = 鈥� \frac{1}{12} [\begin{matrix} 2 + 4^{t} & 4^{t} - 1 \\ 4 - 4^{t} & - 4^{t} - 2 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}]

Hence,

$A^{t} x_{0} = \frac{1}{12} [\begin{matrix} 2 + 4^{t} \\ 4 - 4^{t} \end{matrix}]$

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Most popular questions from this chapter

Find a basis of the linear space Vof all $2 脳 2$ matrices Afor whichrole="math" localid="1659530325801" $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is an eigenvector, and thus determine the dimension of V.

Is $\overset{鈬赌}{v}$ an eigenvector of 7 A? If so, what is the eigenvalue?

Consider the coyotes鈥搑oadrunner system discussed in Example 7. Find closed formulas for c(t) and r(t), for the initial populations $c_{0}$ = 100, $r_{0}$ = 800.

Consider the matrix

$A = [\begin{matrix} 3 & 4 \\ 4 & - 3 \end{matrix}]$

a. Use the geometric interpretation of this transformation as a reflection combined with scaling to find the eigenvaluesA.

b. Find an eigen basis for A.

c. Diagonalize A .

For an arbitrary positive integer n, give a $2 n 脳 2 n$ matrix A without real eigenvalues.

See all solutions

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