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Chapter 7: Q7-19E (page 383)

TRUE OR FALSE

19. If 3is an eigenvalue of an nxn matrix A, then 9 must be an eigenvalue of A2.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

Ax鈫�=位x鈫�, herex鈫�is eigenvector and位is the eigenvalue.

02

Use the eigenvalue formula to find the required value

Consider v is the corresponding vector for the eigenvalue位=3,then

Av=3位

Thus,

A2v=A(3v)=3Av=9v

Therefore,位=9is the eigenvalue of A2.

Hence, the given statement is true.

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Most popular questions from this chapter

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

Give an example of a matrixAof rank 1 that fails to be diagonalizable.

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.

[1234]

IfAis a matrix of rank 1, show that any non-zero vector in the image of Ais an eigenvector of A.

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

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