Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Is $\overset{⇀}{v}$ an eigenvector of $A^{- 1}$ ? If so, what is the eigenvalue?

Yes, the required given value is $\frac{1}{λ}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of eigenvalue

Eigenvalue:An eigenvalue of A is a scalar $λ$ such that the equation $Av = λv$ has a nontrivial solution.

Step 2: Given data

Consider a $n \times n$ invertible matrix A and $\overset{⇀}{v}$ is an eigenvector of the matrix A with respect to the eigenvalue $λ$ .

The objective is to find whether the vector $\overset{⇀}{v}$ is eigenvector for is $A^{- 1}$ or not. If yes find the eigenvalue.

Step 3:Checking whether v⇀an eigenvector of A-1

Since $\overset{⇀}{v}$ is an eigenvector of the matrix A with respect to the eigenvalue $λ$ .

$A \overset{⇀}{v} = λ \overset{⇀}{v}$

Find whether the vector $\overset{⇀}{v}$ is eigenvector for is $A^{- 1}$ or not as,

Consider

role="math" localid="1659528893977" $A \overset{⇀}{v} = λ \overset{⇀}{v} A^{- 1} (A \overset{⇀}{v}) = A^{- 1} (λ \overset{⇀}{v}) A^{- 1} (A^{- 1} A) \overset{⇀}{v} = λ (A^{- 1} \overset{⇀}{v}) I \overset{⇀}{v} = λ (A^{- 1} \overset{⇀}{v}) \sin c e A^{- 1} A = I λ (A^{- 1} \overset{⇀}{v}) = \overset{⇀}{v} A^{- 1} \overset{⇀}{v} = \frac{1}{λ} \overset{⇀}{v}$

Therefore, from equation (1) the vector $\overset{⇀}{v}$ is eigenvector for is $A^{- 1}$ with respect to the eigenvalue $λ^{- 1} = \frac{1}{λ}$ .

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Linear Algebra With Applications

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Short Answer

Is $\overset{⇀}{v}$ an eigenvector of $A^{- 1}$ ? If so, what is the eigenvalue?

Step by Step Solution

Step 1: Definition of eigenvalue

Step 2: Given data

Step 3:Checking whether v⇀an eigenvector of A-1

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Linear Algebra With Applications

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Short Answer

Is v⇀ an eigenvector of A-1? If so, what is the eigenvalue?

Step by Step Solution

Step 1: Definition of eigenvalue

Step 2: Given data

Step 3:Checking whether v⇀an eigenvector of A-1

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Is $\overset{⇀}{v}$ an eigenvector of $A^{- 1}$ ? If so, what is the eigenvalue?