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Chapter 7: Q34E (page 324)

Suppose v鈨�Supposeis an eigenvector of the n脳nmatrix A, with eigenvalue 4 . Explain why v鈨�is an eigenvector of A2+2A+3In.What is the associated eigenvalue?

Short Answer

Expert verified

So, the associated eigen value is 27.

Step by step solution

01

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.

02

Given Data

First ifv鈨�is an Eigenvector of the matrix A with Eigenvalue 4,v鈫� will also be an Eigenvalue for any linear combination of the matrix A.

Thus,v鈨� will be an Eigenvector for as wellA2+2A+3In.

03

Find eigenvalue

Since, we know thatv鈫�is an Eigenvector ofA2+2A+3Inthis means that:

A2+2A+3Inv鈫�=A2v鈫�+2Av鈫�+3lnv鈫�=位2v鈫�+2位v鈫�+3v鈫�=位2+2位+3v鈫�

Since, in this case位=4the associated Eigenvalue forA2+2A+3Inwill be:

位2+2位+3=16+8+3=27

So, the required value is 27.

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Most popular questions from this chapter

Find an eigenbasis for the given matrice and diagonalize:

A=[1111]

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

True or false? If the determinant of a 2 脳 2 matrix A is negative, then A has two distinct real eigenvalues.

A. Find the characteristic polynomial of the matrix

A=[010001abc]

B. Can you find a 3脳3matrix whose characteristic polynomial is-位2+17位2-5位+蟿蟿

Find all2脳2matrices for whiche鈫�1=(10)is an eigenvector with associated eigenvalue 5.
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