Q10E Find all聽2脳2聽matrices for whi... [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 7: Q10E (page 323)

Find all $2 脳 2$ matrices for which $[\begin{matrix} 1 \\ 2 \end{matrix}]$ is an eigenvector with associated eigenvalue 5 .

Short Answer

Expert verified

So, the required matrix is $[\begin{matrix} 5 - & 2 b & b \\ 10 - & 2 d & d \end{matrix}]$ .

Step by step solution

Define the eigenvector

Eigenvector:An eigenvector ofis a nonzero vector vin ${鈩�}_{n}$ such that $A v = 位 v$ , for some scalar $位$ .

Solve for

If $\overset{鈬赌}{v}$ is an Eigen vector of A this means that:

$A \overset{鈬赌}{v} = 位 v$

Now let $A = [\begin{matrix} a & b \\ c & d \end{matrix}], \overset{鈬赌}{v} = [\begin{matrix} 1 \\ 2 \end{matrix}]$ , and $位 = 5$

We want to solve for A.

Evaluation

To do this we will replace everything into the equation $\overset{鈬赌}{A} v = 位 \overset{鈬赌}{v}$ and

We will have:

$A \overset{鈬赌}{v} = 位 \overset{鈬赌}{v} [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 \\ 2 \end{matrix}] = 5 [\begin{matrix} 1 \\ 2 \end{matrix}] [\begin{matrix} a + & 2 b \\ c + & 2 d \end{matrix}] = [\begin{matrix} 5 \\ 10 \end{matrix}] a + 2 b = 5 a = 5 - 2 b$

and

$C + 2 d = 10 c = 10 - 2 d$

Thus, all matrices of the form $[\begin{matrix} 5 - & 2 b & b \\ 10 - & 2 d & d \end{matrix}]$ will satisfy the given requirements.

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91影视

Find all $2 脳 2$ matrices for which $[\begin{matrix} 1 \\ 2 \end{matrix}]$ is an eigenvector with associated eigenvalue 5 .

Short Answer

Step by step solution

Define the eigenvector

Solve for

Evaluation

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91影视

Find all2脳2matrices for which[12]is an eigenvector with associated eigenvalue 5 .

Short Answer

Step by step solution

Define the eigenvector

Solve for

Evaluation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Applied Mathematics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find all $2 脳 2$ matrices for which $[\begin{matrix} 1 \\ 2 \end{matrix}]$ is an eigenvector with associated eigenvalue 5 .