Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find all $2 \times 2$ matrices for which ${\vec{e}}_{1} = (\begin{matrix} 1 \\ 0 \end{matrix})$ is an eigenvector with associated eigenvalue 5.

So, the required matrix is $[\begin{matrix} 5 & b \\ 0 & d \end{matrix}]$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Define the eigenvector

Eigenvector: An eigenvector of A is a nonzero vector v in $R_{n}$ such that $A v = λ v$ , for some scalar $λ$ .

Step 2: Solve for A

If is an Eigen vector of A this means that:

$A \vec{v} = λ^{'} v$

Now let $A = [\begin{matrix} a & b \\ c & d \end{matrix}], \vec{v} = [\begin{matrix} 1 \\ 0 \end{matrix}]$ , and $λ = 5$ .

We want to solve for A.

Step 3: Evaluation

To do this we will replace everything into the equation $A \vec{v} = λ v^{'}$ and we will have:

$\vec{v} = λ \vec{v} [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] = 5 [\begin{matrix} 1 \\ 0 \end{matrix}] [\begin{matrix} a \\ c \end{matrix}] = [\begin{matrix} 5 \\ 0 \end{matrix}] a = 5 c = 0$

b,d are any real number

Thus, all matrices of the form $[\begin{matrix} 5 & b \\ 0 & d \end{matrix}]$ will satisfy the given requirements.

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Linear Algebra With Applications

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Short Answer

Find all $2 \times 2$ matrices for which ${\vec{e}}_{1} = (\begin{matrix} 1 \\ 0 \end{matrix})$ is an eigenvector with associated eigenvalue 5.

Step by Step Solution

Step 1: Define the eigenvector

Step 2: Solve for A

Step 3: Evaluation

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Linear Algebra With Applications

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Find all 2×2 matrices for which e→1=(10) is an eigenvector with associated eigenvalue 5.

Step by Step Solution

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Step 2: Solve for A

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Find all $2 \times 2$ matrices for which ${\vec{e}}_{1} = (\begin{matrix} 1 \\ 0 \end{matrix})$ is an eigenvector with associated eigenvalue 5.