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Chapter 7: Q22E (page 345)

Find all2脳2matrix A for whichE7=鈩�2.

Short Answer

Expert verified

The matrix is of the formA=7007

Step by step solution

01

Algebraic Versus.

Algebraic versus geometric multiplicity If 位 is an eigenvalues of a square matrix A,

thengemu(1)<almu(1)

Let the value of A =A=abcd

We have to find the

A10=70a=7c=0

02

Substitute the values

A01=07b=0d=7A=7007

Therefore, the matrix of the given statement isA=7007

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Most popular questions from this chapter

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.

Prove the part of Theorem 7.2.8 that concerns the trace: If an n 脳 n matrix A has n eigenvalues 位1, . . . , 位n, listed with their algebraic multiplicities, then tr A = 位1+路 路 路+位n.

consider an eigenvalue 位0of ann脳nmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show thatf'(位0)=0(i.e.., the derivative of the characteristic polynomial of A vanishes are位0).

Is v鈬赌an eigenvector ofA-1? If so, what is the eigenvalue?

25: Consider a positive transition matrix

A=[abcd]

meaning that a, b, c, and dare positive numbers such that a+ c= b+ d= 1. (The matrix in Exercise 24 has this form.) Verify that

[bc]and[1-1]

are eigenvectors of A. What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1?

Sketch a phase portrait.
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