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Chapter 1: Q45E (page 40)

Question:IfAandBare two2x2matrices such that the equations Ax鈬赌=0andBx鈫�=0 have the same solutions, then rref (A) must be equal to rref(B) .

Short Answer

Expert verified

The statement, 鈥淚f A and B are two 2x2 matrices such that the equations Ax鈬赌=0andBx鈫�=0 have the same solutions, then rref(A) must be equal to rref(B) .鈥� is true.

Step by step solution

01

Consider the condition

When a matrix is said to be in its reduced row-echelon form, then, it represents the number of solutions present in it.

The number of leading in the reduced row-echelon form of a matrix determines the number of solutions. The number of leading is equal to the number of solutions of the linear system of equations.

As the matrices A and B have same solutions, thus, rref(A) must be equal to rref(B) .鈥�

02

Final answer

The matrices A and B of order 2x2 with equations Ax鈬赌=0andBx鈫�=0 having same solutions must have, .

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Most popular questions from this chapter

Question:Solve the linear system

|y+z=aX+z=bx+y=C|, where a,b andcare arbitrary constants.

a. Using technology, generate a random 3脳4 matrixA . (The entries may be either single-digit integers or numbers between 0and1 , depending on the technology you are using.) Find rref(A). Repeat this experiment a few times.

b. What does the reduced row-echelon form of most3脳4 matrices look like? Explain.

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points (ai,bj)are given to you, and your job is to connect the dots in a reasonably smooth way. Let ai+1>aifori=0,......,n-1.

One method often employed in such design problems is the technique of cubic splines. We choose fi(t), a polynomial of degree 鈮�3, to define the shape of the ride between (ai-1,bi-1)and (ai,bj),fori=0,.....,n.

Obviously, it is required that fi(ai)=biand fi(ai-1)=bi-1,fori=0,.......,n. To guarantee a smooth ride at the points (ai,bi), we want the first and second derivatives of fiand fi+1to agree at these points:

f'i(ai)=f'i+1(ai)and

f''i(ai)=f''i+1(ai),fori=0,.......,n-1.

Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that

f'1(a0)=f'n(an)=0

Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)

If you sell two cows and five sheep and you buy 13 pigs, you gain 1000coins. If you sell three cows and three pigs and buy nine sheep, you break even. If you sell six sheep and eight pigs and you buy five cows, you lose600 coins. What is the price of a cow, a sheep, and a pig, respectively? (Nine Chapters, Chapter 8, Problem 8)

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product two ways: in terms of the columns of A and in terms of the rows of A.

13.

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