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Chapter 7: Q48E (page 384)

If a matrix A has k distinct eigenvalues, then (A)鈮�K

Short Answer

Expert verified

False, for a matrix A has k distinct eigenvalues, thenA鈮�K

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 a matrix A has k distinct eigenvalues:

Consider an example,

A=0011

This matrix is lower-triangular matrix, so its eigenvalues are the entries on its main diagonal, which are 位1=0,位2=1meaning k = 2. But the rows of this matrix are linearly dependent, so rank A = 1 < K. Hence it is false.

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Most popular questions from this chapter

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

0-112

Find an eigenbasis of given matrix and diagonalize it.

A=(1326)

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

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

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

(a)h(0)=f(0)=100(b)h(0)200,f(0)=100(c)h(0)600,f(0)=500

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Reflection about a plane v inR3.

If 0 is an eigenvalue of a matrix A, then det A = 0.
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