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Chapter 7: Q43E (page 338)

Do there existn脳nmatrices A and B such thatAB-BA=lnExplain. Hint: exercise 40 is helpful.

Short Answer

Expert verified

Do exist the matrices.

LHS=AB-BAtr(LHS)=tr(AB-BA)=tr(AB)-tr(BA)

It is never possible to have matrices A and B such that

AB-BA=ln

Step by step solution

01

definition of matrix

A function is defined as a relationship between a set of inputs that each have one output.

Given,

LHS=AB-BAtr(LHS)=tr(AB-BA)tr(AB)-tr(BA)

AS a trace is separable

trLHS=trAB-trAB=0

02

definition of AB-BA=ln

Now ,from exercise 40

=trAB-trBA.

A& B aren脳n matrices.

But in RHS it is In whose trace is always non-zero.

Thus it鈥檚 a contradiction.

It is never possible to have matrices A and B such that

AB-BA=ln

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Most popular questions from this chapter

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 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

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Is v鈬赌an eigenvector ofA+2In? If so, what is the eigenvalue?

If a vector v鈬赌is an eigenvector of both Aand B, isv鈬赌necessarily an eigenvector of A+B?

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.

[2-2001-100003-4002-3]

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

c(t + 1) = 0.75r(t)

r(t + 1) = 鈭�1.5c(t) + 2.25r(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for c(t) and r(t).

(a)c(0)=100,r(0)=200(b)c(0)=r(0)=100(c)c(0)=500,r(0)=700
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