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Chapter 7: Q51E (page 384)

TRUE OR FALSE

If A is a 4 脳 4 matrix with A4 = 0 , then 0 is the only eigenvalue of A.

Short Answer

Expert verified

True, that 0 is the only eigenvalue of A.

Step by step solution

01

Define eigenvalue

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by 位.

02

Explanation for 0 is the only eigenvalue of A

Let 0A be a matrix such that A4 = 0, Let be an eigenvalue of A, with the corresponding eigenvector v 鈮� 0, then Av = 位v.

A4 v = 0

位4v = 0

位 = 0

Thus, 位 = 0 is the only possible eigenvalue of A.

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Most popular questions from this chapter

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.

[04-14]

Consider the matrixA=|1k11|where k is an arbitrary constant. For which values of k does A have two distinct real eigenvalues? When is there no real eigenvalue?

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Is v鈬赌an eigenvector ofA+2In? If so, what is the eigenvalue?

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