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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: Q35E (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. Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.
$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

Short Answer

Expert verified

$A^{1001} = A$ , and the matrix A describes a reflection of the diagonal line $y = x$ .

Step by step solution

01

Calculate

Let

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

Then we have

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

Therefore, we conclude that

$A^{n} = I_{2}$ , if n is even and $A^{n} = A$ if is odd.

Hence

$A^{1001} = A$

02

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

Since power $A^{n}$ alternates between A and $I_{2}$ and

$A = [\begin{matrix} c o s 90^{o} & s i n 90^{o} \\ s i n 90^{o} & - c o s 90^{o} \end{matrix}]$

So the matrix describes a reflection about the line which is passing through origin and whose slope is $45 掳$ .

03

The Final Answer

Therefore, A describes a reflection of the diagonal line $y = x$ .

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Most popular questions from this chapter

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.

In the financial pages of a newspaper, one can sometimes find a table (or matrix) listing the exchange rates between currencies. In this exercise we will consider a miniature version of such a table, involving only the Canadian dollar $(C \()$ and the South African Rand $(Z A R)$ . Consider the matrix

role="math" localid="1659786495324" $\begin{array}{rcl} \begin{matrix} C\) & ZAR \end{matrix} \\ A & = & [\begin{matrix} 1 & 1 / 8 \\ 8 & 1 \end{matrix}] \begin{matrix} C \( \\ ZAR \end{matrix} \end{array}$

representing the fact thatrole="math" localid="1659786520551" $C \) 1$ is worth role="math" localid="1659786525050" $Z A R 8$ (as of September 2012).

a. After a trip you have $C $ 100$ and $Z A R 1, 600$ in your pocket. We represent these two values in the vector $\overset{鈫�}{x} = [\begin{matrix} 100 \\ 1, 600 \end{matrix}]$ . Compute $A \overset{鈫�}{x}$ . What is the practical significance of the two components of the vector $A \overset{鈫�}{x}$ ?

b. Verify that matrix $A$ fails to be invertible. For which vectors $\overset{鈫�}{b}$ is the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ consistent? What is the practical significance of your answer? If the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent, how many solutions $\overset{鈫�}{x}$ are there? Again, what is the practical significance of the answer?

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix $[\begin{matrix} 0.4 & 0.3 \\ 0.6 & 0.7 \end{matrix}] a n d \overset{炉}{x}$ is the distribution vector $[\begin{matrix} 1 \\ 0 \end{matrix}]$ . [We will not be using parts (a) and (c) of Theorem 2.3.11]. The general proof of Theorem 2.3.11 runs along similar lines, as we will see in Chapter 7.

Compute $A [\begin{matrix} 1 \\ 2 \end{matrix}] a n d A [\begin{matrix} 1 \\ - 1 \end{matrix}]$ . Write $A [\begin{matrix} 1 \\ - 1 \end{matrix}]$ as a scalar multiple of the vector $[\begin{matrix} 1 \\ - 1 \end{matrix}]$ .
Write the distribution vector $\overset{鈫�}{x} = [\begin{matrix} 1 \\ 0 \end{matrix}]$ as a linear combination of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] a n d [\begin{matrix} 1 \\ - 1 \end{matrix}]$
Use your answers in part (a) and (b) to write $A \overset{鈫�}{x}$ as a linear combination of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] a n d [\begin{matrix} 1 \\ - 1 \end{matrix}]$ . More generally, write $A^{m} \overset{鈫�}{x}$ as a linear combination of vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] a n d [\begin{matrix} 1 \\ - 1 \end{matrix}]$ , for any positive integer m. See Exercise 81.
In your equation in part (c), let got to infinity to find $\underset{m 鈫� 鈭�}{l i m} A^{m} \overset{鈫�}{x}$ . Verify that your answer is the equilibrium distribution for A.

There exists an invertible 2 脳 2 matrix A such that $A^{- 1} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises16through 23 . Show the effect of these transformations on the letter $L$ considered in Example 5. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

22. $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

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