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Chapter 7: Q48E (page 347)

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

[00010a010]

Short Answer

Expert verified

The matrix is A diagonalizable, only if the values of a > 0

Step by step solution

01

Calculating the value of of the matrix .

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

The given matrix is00010a010

The characteristic equation is,

-位位2-a=0=位=0,卤a

The value must be a鈮�0.

02

Analyzing whether the matrix A is diagonalizable.

For a>0, the matrix A has three distinct eigenvalues and by theorem 7.3.4 , the matrix is diagonalizable.

For a = 0, note that dim(kerA)鈮�3. Hence, the matrix A is not diagonalizable.

So, the matrix A is diagonalizable only if a > 0

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Most popular questions from this chapter

find an eigenbasis for the given matrice and diagonalize:

A=114[123246369]

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.

[-1-1-1-1-1-1-1-1-1]

In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which [12]is an eigenvector.

(a) Find a basis of V and thus determine the dimension of V.

(b) Consider the linear transformation T (A) = A[12] from V to R2. Find a basis of the image of Tand a basis of the kernel of T. Determine the rank of T .

(c) Consider the linear transformation L(A) = A[13] from V to R2. Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.

if A is a 2脳2matrix with t r A = 5and det A = - 14what are the eigenvalues of 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

5.(45-2-2)
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