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Chapter 7: Q45E (page 346)

For which values of constants a,b,c are the matrix diagonalizable?

[1ab02c003]

Short Answer

Expert verified

The matrix A is diagonalizable for all values ofa,b,c鈭�鈩�3

Step by step solution

01

Calculating the Characteristic Equation of the matrix .

Consider an n脳nmatrix A and a scalar 位. Then 位is the eigenvalue of A if det(A-位濒)=0. This is called Characteristic equation of matrix .

The given matrix is1ab02c003

This matrix has an upper triangular shape. As a result, the entries on its main diagonal are its eigenvalues.

They are位1=1,位2=2,位3=3

02

Analyzing whether the matrix A is diagonalizable.

Because a 3脳3matrix has three distinct eigenvalues, the matrix A is diagonalizable for all values ofrole="math" localid="1659593329128" a,b,c鈭�鈩�

Finally we can conclude that A is diagonalizable for alla,b,c鈭�鈩�

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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.

[51-521082-7]

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

x炉(t+1)=[0.50.250.50.75]x炉(t)

with initial value x0鈫�=e1鈫�. Then do the same for the initial value x0鈫�=e2鈫�. Sketch the two trajectories.

b. Consider the matrix

A=[0.50.250.50.75]

.

Using technology, compute some powers of the matrix A, say, A2, A5, A10, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If A=[abcd]

is an arbitrary positive transition matrix, what can you say about the powers Atas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 脳 2.

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

14.(100-500001)

Question: If a vector v鈫�is an eigenvector of both An脳n, Awith associated eigenvalue 位, what can you say aboutker(A-位滨n)?Is the matrixA-位滨ninvertible?

Give an example of a matrixAof rank 1 that fails to be diagonalizable.
See all solutions

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