Solution: In Exercises 7–10, the augmented matrix of a

Problem 2E Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. 3x1 + 6x2 = –3 5x1 + 7x2 = 10

Problem 5E Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

Problem 4E Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. Find the point of intersection of the lines x1 + 2x2 = – 13 and 3x1 – 2x2 = 1

Problem 34E An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let T1,…,T4 denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below.3 For instance, T 1 = (10 + 20 T2 + T4)/4, or 4T1 – T2 – T4 = 30 Solve the system of equations from Exercise 33. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting “replace” operations.] Exercise 33: Write a system of four equations whose solution gives estimates for the temperatures T1,…,T4.

Problem 1E Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. x1 + 5x2 = 7 –2x1 – 7x2 = –5

Problem 3E Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. Find the point (x1, x2) that lies on the line x1 + 2x2 = 4 and on the line x1 – x2 = 1. See the figure.

Problem 6E Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

Problem 8E In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

Problem 9E In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

Problem 10E In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

Problem 15E Determine if the systems in Exercises 15 – 16 are consistent. Do not completely solve the systems.

Problem 13E Solve the systems in Exercises 11–14. x1 –3x3 = 8 2x1 + 2x2 + 9x3 = 7 x2 + 5x3 = –2

Problem 16E Determine if the systems in Exercises 15 – 16 are consistent. Do not completely solve the systems. 2x1 – 4x4 = – 10 3x2 + 3x3 = 0 x3 + 4x4= – 1 – 3x1 + 2x2 + 3x3 + x4 = 5

Problem 12E Solve the systems in Exercises 11–14. x1 – 5x2 + 4x3 = –3 2x1 – 7x2 + 3x3 = –2 – 2x1 + x2 + 7x3 = –1

Problem 11E Solve the systems in Exercises 11–14. x2 + 5x3 = –4 x1 + 4x2 + 3x3 = –2 2x1 + 7x2 + x3 = –2

Problem 14E Solve the systems in Exercises 11–14. 2x1 –6x3 = –8 x2 + 2x3 = 3 3x1 + 6x2 – 2x3 = –4

Problem 17E Do the three lines 2x1 + 3x2 = –1, 6x1 + 5x2 = 0 and 2x1 – 5x2 = 7 have a common point of intersection? Explain.

Problem 18E Do the three planes 2x1 + 4x2 + 4x3 = 4, x2 – 2x3 = – 2 and 2x1 + 3x2 = 0 have at least one common point of intersection? Explain

Problem 19E In Exercises 19–22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 21E In Exercises 19–22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 20E In Exercises 19–22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 22E In Exercises 19–22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 24E In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text. a. Two matrices are row equivalent if they have the same number of rows. b. Elementary row operations on an augmented matrix never change the solution set of the associated linear system. c. Two equivalent linear systems can have different solution sets. d. A consistent system of linear equations has one or more solutions.

Problem 23E In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text. a. Every elementary row operation is reversible. b. A 5 × 6 matrix has six rows. c. The solution set of a linear system involving variables x1,...,xn is a list of numbers.(s1,…,sn) that makes each equation in the system a true statement when the values s1,…,sn are substituted for x1,...,xn respectively. d. Two fundamental questions about a linear system involve existence and uniqueness.

Problem 25E Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system:

Problem 26E Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer. 2x1 + 4x2 = f cx1 + dx2 = g

Problem 27E Suppose a, b, c, and d are constants such that a is not zero and the system below is consistent for all possible values of f and g. What can you say about the numbers a, b, c, and d? Justify your answer. ax1 + bx2 = f cx1 + dx2 = g

Problem 28E Construct three different augmented matrices for linear systems whose solution set is x1= 3, x2 = –2, x3 = –1

Problem 29E In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

Problem 30E In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

Problem 31E In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

Problem 32E In Exercises 29–32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

Problem 33E An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let T1,…,T4 denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below.3 For instance, T 1 = (10 + 20 T2 + T4)/4, or 4T1 – T2 – T4 = 30 Write a system of four equations whose solution gives estimates for the temperatures T1,…,T4.