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Chapter 7: Q42E (page 346)

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

A=[1a0b]

Short Answer

Expert verified

The matrixa,b鈭�鈩�x鈩�\1U0,1

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:

detA-位濒=01-位a0b-位=01-位b-位=0位1=1,位2=b

For b鈮�1, using Ex. 41, we conclude that A is diagonalizable for arbitrary a. For b = 1, though, using Ex. 40, we conclude that A is only diagonalizable for a=0.

Here, the final result is a,b鈭�鈩�x鈩�\1U0,1.

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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.

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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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find an eigenbasis for the given matrice and diagonalize:

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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.

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