Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.
$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Eigenvalues are: $λ_{1} = \frac{5 + \sqrt{33}}{2}$ with multiplicity of 1

with multiplicity of 1

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by $λ$ , is the factor by which the eigenvector is scaled.
Eigenvalues of a triangular matrix are its diagonal matrix.

Step 2: Finding all real eigenvalues, with their algebraic multiplicities

We can clearly see that,

$\det ({λI}_{n}) = \det [\begin{matrix} 1 - λ & 2 \\ 3 & 4 - λ \end{matrix}] = (1 - λ) (4 - λ) - 2.3 = λ^{2} - 5 λ - 2$

Find zeroes of characteristic equation:

role="math" localid="1659585122780" $λ^{2} - 5 λ - 2 = 0 λ = \frac{5 + \sqrt{25 + 8}}{2} λ = \frac{5 + \sqrt{33}}{2}$

Eigenvalues are:

$λ_{1} = \frac{5 + \sqrt{33}}{2}$ with multiplicity of 1

$λ_{2} = \frac{5 + \sqrt{33}}{2}$ with multiplicity of 1

Hence, the answer is $λ = \frac{5 + \sqrt{33}}{2}$ .

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Linear Algebra With Applications

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Short Answer

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.
$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Step by Step Solution

Step 1: Eigenvalues

Step 2: Finding all real eigenvalues, with their algebraic multiplicities

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Linear Algebra With Applications

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Short Answer

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.[1234]

Step by Step Solution

Step 1: Eigenvalues

Step 2: Finding all real eigenvalues, with their algebraic multiplicities

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Pure Maths

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.
$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$