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.
$[\begin{matrix} 2 & - 2 & 0 & 0 \\ 1 & - 1 & 0 & 0 \\ 0 & 0 & 3 & - 4 \\ 0 & 0 & 2 & - 3 \end{matrix}]$

Eigenvalues are:

$\begin{array}{l} λ_{1} = 0, almuu (0) = 1 \\ λ_{2} = - 1, almu (- 1) = 2 \\ λ_{4} = 1, almu (1) = 1 \end{array}$

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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

Since, given matrix is triangular its eigenvalues are the entries on the main diagonal.

Find eigenvalues using characteristic equation as:

$d e t (A - λ l) = 0 |\begin{matrix} 2 - λ & - 2 & 0 & 0 \\ 1 & - 1 - λ & 0 & 0 \\ 0 & 0 & 3 - λ & - 4 \\ 0 & 0 & 2 & - 3 \end{matrix}| = 0 ((2 - λ) (- 1 - λ) + 2) ((3 - λ) (- 3 - λ) + 8) = 0$

$λ = 0, λ_{2, 3} = - 1, λ_{4} = 1$

Hence, the answer is:

$\begin{array}{l} λ_{1} = 0, almuu (0) = 1 \\ λ_{2} = - 1, almu (- 1) = 2 \\ λ_{4} = 1, almu (1) = 1 \end{array}$

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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.
$[\begin{matrix} 2 & - 2 & 0 & 0 \\ 1 & - 1 & 0 & 0 \\ 0 & 0 & 3 & - 4 \\ 0 & 0 & 2 & - 3 \end{matrix}]$

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.[2-2001-100003-4002-3]

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.
$[\begin{matrix} 2 & - 2 & 0 & 0 \\ 1 & - 1 & 0 & 0 \\ 0 & 0 & 3 & - 4 \\ 0 & 0 & 2 & - 3 \end{matrix}]$