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Chapter 2: Q40 E (page 86)

For the matrix A in exercises 33 through 42 , compute, A2=AA, A3=AAAand 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=0-110

Short Answer

Expert verified

A1001=0-110

And the matrix A describes a rotation by90o=蟺2 in the counter clockwise direction.

Step by step solution

01

Calculate

Let

A=0-111

Then we have

role="math" localid="1664205846758" A2=AAA=0-1110-111=-100-1=-I2

A3=A2A=-I2A=-AA4=A3A=-AA=-A2=I2

Since, A4=I2, this gives us A1000=I2, as 1000=250脳4, Hence we get

A1001=A1000A=I2A=A=0-111

02

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

Clearly,

A=cos蟺/2sin蟺/2-sin蟺/2cos蟺/2

So the matrix A describes a rotation by 90掳in the counter clockwise direction.

03

The Final Answer

Hence,

A1001=0-110

And the matrix A describes a rotation by 90o=蟺2in the counter clockwise direction.

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Most popular questions from this chapter

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix [0.40.30.60.7]andx炉is the distribution vector[10]. [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.

  1. ComputeA[12]andA[1-1]. WriteA[1-1]as a scalar multiple of the vector[1-1].
  2. Write the distribution vectorx鈫�=[10]as a linear combination of the vectors[12]and[1-1]
  3. Use your answers in part (a) and (b) to writeAx鈫�as a linear combination of the vectors[12]and[1-1]. More generally, write Amx鈫�as a linear combination of vectors[12]and[1-1], for any positive integer m. See Exercise 81.
  4. In your equation in part (c), let got to infinity to find limm鈫�鈭�Amx鈫�. Verify that your answer is the equilibrium distribution for A.

Find the matrices of the transformations Tand Ldefined in Exercise 80.

For which values of the constant k is the following matrix invertible?

[11112k14k2]

TRUE OR FALSE?

If A is an invertible2脳2 matrix and B is any 2脳2 matrix, then the formula rref (AB) = rref (B)must hold.

Question: Consider the matrix A2 in Example 4 of Section 2.3.

  1. The third component of the first column ofA2 is 1/4 . What does this entry mean in practical terms, that is, in terms of surfers following links in our mini-Web?
  2. When is the i j thentry of A2 equal to 0 ? Give your answer both in terms of paths of length 2 in the graph of the mini-Web and also in terms of surfers being able to get from page j to pageiby following two consecutive links.
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