Q39 E For the matrix A in exercise 33 ... [FREE SOLUTION]

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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: Q39 E (page 86)

For the matrix A in exercise 33 through 42 , Compute $A^{2} = A A$ , and localid="1664204847004" $A^{3} = A A A$ and localid="1664205252603" $A^{4}$ . 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.
$A = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}]$

Short Answer

Expert verified

$A^{100} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}]$

And the matrix describes a rotation by $45^{o} = \frac{蟺}{4}$ in the clockwise direction.

Step by step solution

Calculate

Let

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

Then we have

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

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

Since, $A^{4} = - I_{2}$ , this gives us $A^{8} = A^{4} A^{4} = I_{2}$ . Since $1000 = 125 脳 8,$ Hence we get

$A^{1001} = A^{1000} A = I_{2} A = A = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}]$

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

$A = [\begin{matrix} c o s (蟺 / 4) & s i n (蟺 / 4) \\ - s i n (蟺 / 4) & \cos (蟺 / 4) \end{matrix}]$

So the matrix describes a rotation $45^{o} = \frac{蟺}{4}$ in the clockwise direction.

Final Answer

Therefore,

$A^{100} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}]$

And the matrix describes a rotation by $45^{o} = \frac{蟺}{4}$ in the clockwise direction.

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Short Answer

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

Final Answer

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