Q37E For the matrix A in exercise 33 ... [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 2: Q37E (page 86)

For the matrix A in exercise 33 through 42 , Compute $A^{2} = A A, A^{3} = A A A$ , and $A^{4}$ . Describe the pattern that emerges, and use this pattern to find $A^{1001}$ . Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.
$A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

Short Answer

Expert verified

$A^{1001} = [\begin{matrix} 1 & 0 \\ - 1001 & 1 \end{matrix}]$ and the matrix describes a reflection about the axis.

Step by step solution

Calculate

Let

$A = [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}]$

Then we have

$A^{2} = A A = [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}]$

$A^{3} = A^{2} A = [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ - 3 & 1 \end{matrix}] A^{4} = A^{3} A = [\begin{matrix} 1 & 0 \\ - 3 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ - 4 & 1 \end{matrix}]$

Therefore, we conclude that

$A^{n} = [\begin{matrix} 1 & 0 \\ - n & 1 \end{matrix}]$

$A^{1001} = [\begin{matrix} 1 & 0 \\ - 1001 & 1 \end{matrix}]$

Representation of matrix A in terms of rotations, reflections, and projections

Clearly, the matrix A and its power $A^{n}$ represents a vertical shear along the y-axis. For, A the shear magnitude is 1 and the shear magnitude for power $A^{n}$ is n. That means, the shear magnitude increases linearly in n .

The Final Answer

Therefore,

$A^{1001} = [\begin{matrix} 1 & 0 \\ - 1001 & 1 \end{matrix}]$ and the matrix describes a reflection about the axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Calculate

Representation of matrix A in terms of rotations, reflections, and projections

The Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

91影视

For the matrix A in exercise 33 through 42 , Compute A2=AA,A3=AAA, and A4. Describe the pattern that emerges, and use this pattern to find A1001. Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.A=100-1

Short Answer

Step by step solution

Calculate

Representation of matrix A in terms of rotations, reflections, and projections

The Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.