Q48E Question: In Exercises 43 throug... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q48E (page 86)

Question: In Exercises 43 through 48, find a 2x2 matrix A with the given properties. Hint: It helps to think of geometrical examples.
10. $A^{10} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Short Answer

Expert verified

Answer:

Hence, the required matrix is $A = [\begin{matrix} 0 & \frac{1}{10} \\ 0 & 1 \end{matrix}]$ .

Step by step solution

Linear Transformation

For any matrix of the form, $B = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ , the general form of the matrix is given by $B^{k} = [\begin{matrix} 1 & k \\ 0 & 1 \end{matrix}]$ .

Find the matrix

Given, a matrix $A^{10} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ . Let there be any matrix $C = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ .

Then,

$C^{2} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 + 0 & 1 + 1 \\ 0 + 0 & 0 + 1 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$

Similarly,

$C^{3} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 + 0 & 2 + 1 \\ 0 + 0 & 0 + 1 \end{matrix}] = [\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}]$

So, it is clear that, the matrix Cⁿ will result gives $[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}]$ as a solution.

Now, in order to have $A^{10} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ , we must have a matrix A such that $A = [\begin{matrix} 0 & \frac{1}{10} \\ 0 & 1 \end{matrix}]$ which gives:

$A^{10} = [\begin{matrix} 1 & 10 脳 \frac{1}{10} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Hence, the required matrix is $A = [\begin{matrix} 0 & \frac{1}{10} \\ 0 & 1 \end{matrix}]$ .

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91影视

Question: In Exercises 43 through 48, find a 2x2 matrix A with the given properties. Hint: It helps to think of geometrical examples.
10. $A^{10} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Short Answer

Step by step solution

Linear Transformation

Find the matrix

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91影视

Question: In Exercises 43 through 48, find a 2x2 matrix A with the given properties. Hint: It helps to think of geometrical examples.10. A10=[1101]

Short Answer

Step by step solution

Linear Transformation

Find the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Question: In Exercises 43 through 48, find a 2x2 matrix A with the given properties. Hint: It helps to think of geometrical examples.
10. $A^{10} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$