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Chapter 7: Q72E (page 360)

Use the method outlined in Exercise 70 to check for which values of the constants a, b, and c the matrix A=[1ab00c001]is diagonalizable.

Short Answer

Expert verified

The given matrix A is diagonaziable when b=-ac.

Step by step solution

01

Definition

A scalar associated with a given linear transformation of a vector space and having the property that there is some nonzero vector which when multiplied by the scalar is equal to the vector obtained by letting the transformation operate on the vector especially: a root of the characteristic equation of a matrix.

02

Solving the matrix

The matrix A is upper triangular, so its eigenvalues are the entries on its main diagonal, which are位1=0位2,3=1.

Now, we have

A-0l=1ab00c001

A-l=0ab0-1c000

Multiply both the obtained matrices,

AA-l=00b+aC000000

So, for A is diagonalizable, we need, b+ac=0

鈬�b=-ac

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Most popular questions from this chapter

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

If v鈬赌is an eigenvector of matrix A, show that is in the image of A.or in the kernel ofA.

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.

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

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]

Is v鈬赌an eigenvector of 7 A? If so, what is the eigenvalue?
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