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Chapter 2: Q62E (page 109)

TRUE OR FALSE?

For every transition matrix A there exists a nonzero vector x鈫� such thatAx鈫�=x鈫�.

Short Answer

Expert verified

The statement is false.

Step by step solution

01

Consider the parameters.

Consider the system.

Ax=x

02

Consider the condition.

The system is Ax=ximplies,

Ax=x鈬�Ax-x=0鈬�(A-l)x=0

Let any matrix B=A-lthen it is written as,

鈬�Bx=0

This implies, Bx=0鈮�Ax=x.

Thus, the system will have infinitely many solutions.

Hence, the statement is false.

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Most popular questions from this chapter

Some parking meters in downtown Geneva, Switzerland, accept2Franc and5 Franc coins.

a. A parking officer collects 51 coins worth 144Francs. How many coins are there of each kind?

b. Find the matrixAthat transforms the vector

[number of 2 Franc coinsnumber of5Franc coins]

into the vector

[total value of coinstotalnumberof coins]

c. Is the matrixAin part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

If A2 is invertible, then matrix A itself must be invertible.

Consider a linear transformation Tfrom R2to R2. Suppose thatv鈫�and w鈫� are two arbitrary vectors in R2and thatx鈫�is a third vector whose endpoint is on the line segment connecting the endpoints ofv鈫�and w鈫�. Is the endpoint of the vectorT(x)鈫�necessarily on the line segment connecting the endpoints ofT(v)鈫�and T(w)鈫�? Justify your answer.

Matrixis invertible for all real numbers k.

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T(x)鈫�=Ax鈫�on this face.

24. [0-110]
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