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Chapter 7: Q49E (page 346)

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?

A =[00a100010]

Short Answer

Expert verified

The A=00a100010matrix is not diagonalizable

Step by step solution

01

Characteristic equation 

Consider an n 脳n matrix A and a scalar 位. Then 位 is an eigenvalue5 of A.

det(A-位濒)=0

This is called the characteristic equation (or the secular equation) of matrix A.

02

Solution of the problem

We solve:

det(A-位l)=0-位0a1-位001-位=0-位3+a=0

The only real root of this equation is 位=a13, thus we don't have three real eigenvalues (counting their algebraic multiplicities), So A is not diagonalizable.

Therefore, the final result A is not diagonalizable.

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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

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25: Consider a positive transition matrix

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e鈫�=11..1

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For each of the matrices A in Exercise1 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

[1111]

Find a basis of the linear space Vof all3脳3matrices Afor which both[100]and[001]are eigenvectors, and thus determine the dimension of.
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