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Chapter 7: Q16E (page 336)

Consider the matrix A=|abbc| where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

Short Answer

Expert verified

Values of a, b, and c for which A have two distinct eigenvaluesa鈮�c,b鈮�0

Step by step solution

01

Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.

02

Step 2: Finding values of a, b, and c for which A have two distinct eigenvalues: 

We can clearly see that,

detA-位l=0deta-位bbc-位=0a-位c-位-b2=0位2-a+cc-位-b2=0位1,2=a+c卤a2+2ac+c2-4ac+b22=a+c卤a-c2+b22

Hence, a鈮�c,b鈮�0.

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Most popular questions from this chapter

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

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

Find an eigenbasis of given matrix and diagonalize it.

A=(1326)

Consider an upper triangular n脳nmatrix Awithaij鈮�0fori=1,2,鈥�,mandaij=0fori=m+1,鈥�n. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

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

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