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.
$[\begin{matrix} - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \end{matrix}]$

Eigenvalues are:

$λ_{1, 2} = 0, almu (0) = 2 λ_{3} = 3, almu (3) = 3$

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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,

$\det (A - λ) = 0 |\begin{matrix} - 1 - λ & - 1 & - 1 \\ - 1 & - 1 - λ & - 1 \\ - 1 & - 1 & - 1 - λ \end{matrix}| = 0 {(1 - λ)}^{3} - 1 - 1 - 3 (- 1 - λ) = 0 λ^{2} (λ - 3) = 0 λ_{1, 2} = 0, λ_{3} = 3$

Hence, the answer is:

$λ_{1, 2} = 0, almu (0) = 2 λ_{3} = 3, almu (0) = 3$

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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.
$[\begin{matrix} - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \end{matrix}]$

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.[-1-1-1-1-1-1-1-1-1]

Step by Step Solution

Step 1: Eigenvalues

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

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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.
$[\begin{matrix} - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \end{matrix}]$