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Q38E

Expert-verified

Linear Algebra With Applications

Found in: Page 324

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

We are told that $[\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}]$ is an eigenvector of the matrix $[\begin{matrix} 4 & 1 & 1 \\ - 5 & 0 & - 3 \\ - 1 & - 1 & 2 \end{matrix}]$ what is the associated eigenvalue?

Hence, the required eigenvalue is $λ = 2$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

Step 2: Finding eigenvalues

Consider $x = [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}]$ be an eigen vector of the matrix $[\begin{matrix} 4 & 1 & 1 \\ - 5 & 0 & - 3 \\ - 1 & - 1 & 2 \end{matrix}]$ .

The objective is to find the associated eigenvalue.

If x is an eigen vector of A then $A x = λ x$ where scalar $λ$ is called an eigenvalue of A.

Consider,

$A x = λ x [\begin{matrix} 4 & 1 & 1 \\ - 5 & 0 & - 3 \\ - 1 & - 1 & 2 \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}] = λ [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}] [\begin{matrix} 4 & 1 & 1 \\ - 5 & 0 & - 3 \\ - 1 & - 1 & 2 \end{matrix}] = λ [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}] [\begin{matrix} 2 \\ - 2 \\ - 2 \end{matrix}] = λ [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}] 2 [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}] = λ [\begin{matrix} 1 \\ - 1 \\ - 1 \end{matrix}]$

Hence, the associated eigen value is $λ = 2$

Most popular questions for Math Textbooks

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

c(t + 1) = 0.75r(t)

r(t + 1) = −1.5c(t) + 2.25r(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for c(t) and r(t).

$(a) c (0) = 100, r (0) = 200 (b) c (0) = r (0) = 100 (c) c (0) = 500, r (0) = 700$

Suppose $v ⃗$ Suppose is an eigenvector of the $n \times n$ matrix A, with eigenvalue 4 . Explain why $v ⃗$ is an eigenvector of $A^{2} + 2 A + 3 I_{n} .$ What is the associated eigenvalue?

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise 1 through 20 ,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

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

Find a $2 \times 2$ matrix A such that $[\begin{matrix} 3 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ are eigenvectors of A , with eigenvalues 5 and 10 , respectively.

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

$5 . (\begin{matrix} 4 & 5 \\ - 2 & - 2 \end{matrix})$

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