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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 2: Q33 E (page 86)

For the matrix A, compute, $A^{2} = A A, A^{3} = A A A$ and $A^{4}$ . Describe the pattern that emerges, and use this pattern to find it $A^{1001}$ . 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

$A^{1001} = A$ , and the matrix A represents a rotation by $180^{o}$ .

Step by step solution

01

Calculate

Let

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

Then we have

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

Now,

$A^{3} = A^{2} A = I_{2} A = A A^{4} = A^{3} A = A A = A^{2} = I_{2}$

Therefore, we conclude that $A^{n} = I_{2}, i f n i s e v e n a n d A^{n} = A i f n i s o d d .$

Hence,

$A^{1001} = A$

02

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

Since power $A^{n}$ alternates between A and $I_{2} = - I_{2}$ , so the matrix A describes a reflection of the origin. Also, notice that

$A = [\begin{matrix} \cos 蟺 & \sin 蟺 \\ - \sin 蟺 & \cos 蟺 \end{matrix}]$

Thus alternatively one can say represents a rotation by $180 掳$

03

The Final Answer

Therefore,

$A^{1001} = A$ , and the matrix represents a rotation by $180^{o}$ .

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Most popular questions from this chapter

TRUE OR FALSE?

For every transition matrix A there exists a nonzero vector $\overset{鈫�}{x}$ such that $A \overset{鈫�}{x} = \overset{鈫�}{x}$ .

A nonzero matrix of the form $A = [\begin{matrix} a & - b \\ b & a \end{matrix}]$ represents a rotation combined with ascaling. Use the formula derived in Exercise 2.1.13 to find the inverse of A .Interpret the linear transformation defined by $A^{- 1}$ geometrically. Explain.

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?21 $f (x) = x^{2}$ .

Consider the transformation T from $R^{2}$ to $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through an angle of $45 掳$ in the counterclockwise direction, as shown in the following figure:

You are told that T is a linear transformation. (This will be shown in the next section.) Find the matrix of T .

TRUE OR FALSE?

If a matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ represents the orthogonal projection onto a line L , then the equation $a^{2} + b^{2} + c^{2} + d^{2} = 1$ must hold.

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