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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 1: Q38 E (page 1)

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.
$A = \frac{1}{2} [\begin{matrix} - 1 & - \sqrt{3} \\ \sqrt{3} & - 1 \end{matrix}]$

Short Answer

Expert verified

$A^{1001} = \frac{1}{2} [\begin{matrix} - 1 & \sqrt{3} \\ - \sqrt{3} & - 1 \end{matrix}]$

And the matrix describes a rotation through angle $120 掳$ in the counter clockwise direction.

Step by step solution

01

Calculate

Let

$A = \frac{1}{2} [\begin{matrix} - 1 & \sqrt{3} \\ - \sqrt{3} & - 1 \end{matrix}]$

Then we have

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

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

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

Clearly,, $A^{999} = I_{2}$ as $999 = 333 脳 3$ and $A^{3} = I_{2}$ .

Hence we get

$A^{1001} = A^{999} A^{2} = A^{2} = \frac{1}{2} [\begin{matrix} - 1 & \sqrt{3} \\ - \sqrt{3} & - 1 \end{matrix}]$

02

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

Clearly,

A = [\begin{matrix} c o s (2 蟺 / 3) & - \sin (2 蟺 / 3) \\ \sin (2 蟺 / 3) & c o s (2 蟺 / 3) \end{matrix}]

So the matrix A describes a rotation by $120 掳 = \frac{2 蟺}{3}$ in the counter clockwise direction.

03

The Final Answer

Therefore,

$A^{1001} = \frac{1}{2} [\begin{matrix} - 1 & \sqrt{3} \\ - \sqrt{3} & - 1 \end{matrix}]$

And the matrix describes a rotation through angle $120 掳$ in the counter clockwise direction.

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Most popular questions from this chapter

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product two ways: in terms of the columns of A and in terms of the rows of A.

13.

$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 7 \\ 11 \end{matrix}]$

If the positive definite matrix Ais similar to the symmetric matrix B, then Bmust be positive definite as well.

The reduced row-echelon forms of the augmented matrices of three systems are given here. How many solutions does each system have?

$a . [\begin{matrix} 1 & 0 & 2 & | \\ 0 & 1 & 3 & | \\ 0 & 0 & 0 & | \end{matrix} \begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] b . [\begin{matrix} 1 & 0 & | & 5 \\ 0 & 1 & | & 6 \end{matrix}] c . [\begin{matrix} 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & 3 \end{matrix}]$

If Ais a symmetric nxnmatrix such that $A^{n} = 0$ , then Amust be the zero matrix.

a. Using technology, generate a random $4 脳 3$ matrix A. (The entries may be either single-digit integers or numbers between 0 and 1, depending on the technology you are using.) Find $rref (A)$ . Repeat this experiment a few times.

b. What does the reduced row-echelon form of most $4 脳 3$ matrices look like? Explain.

See all solutions

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