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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

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

Find a basis of the linear space Vof all2脳2matrices Afor which[1-3] is an eigenvector, and thus determine the dimension of V.

(a). If 2i is an eigenvalue of a real 2 脳 2 matrix A, findA2.

(b). Give an example of a real 2 脳 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. ComputeA2and check that your answer agrees with part (a).

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

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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