Section 6.1 Matrices and Systems of Equations

Before Starting this Section, Review

  1. 1 Systems of equations (Sections 5.1 and 5.2)

  2. 2 Equivalent systems (Section 5.2 , page 528)

  3. 3 Gaussian elimination (Section 5.2 , page 529)

Objectives

  1. 1 Define a matrix.

  2. 2 Use matrices to solve a system of linear equations.

  3. 3 Use Gaussian elimination to solve a system.

  4. 4 Use Gauss–Jordan elimination to solve a system.

Leontief Input–Output Model

In 1949, the Russian-born Harvard Professor Wassily Leontief opened the door to a new era in mathematical modeling in economics. He divided the U.S. economy into 500 “sectors,” such as the coal, automotive, and communications industries and agriculture. For each sector, he wrote a linear equation that described how that sector distributed its output to other sectors of the economy. In 1949, the largest computer was the Mark II, and it could not handle the resulting system of 500 equations in 500 variables. Leontief then combined some of the sectors and reduced the system to 42 equations in 42 variables to solve the problem. As computers have become more powerful, scientists and engineers can now work on problems far more complex than they ever dreamed of a few decades ago. In Example 10, we explore Leontief’s input–output model applied to a simple economy.

Definition of a Matrix

  1. 1 Define a matrix.

Suppose Great Builders Incorporated (GBI), builds ranch, colonial, and modern houses. GBI wants to compare the cost of labor and the cost of material involved in building each type of house during one year. These data could be represented as in Table 6.1.

Table 6.1

Type of House Ranch Colonial Modern
Cost of labor (in millions) 12 13 14
Cost of material (in millions) 9 11 10

Or it could be represented more compactly by the following array:

[12131491110].

This rectangular array of numbers is called a matrix (plural, matrices). This matrix has two rows (the types of costs) and three columns (the types of houses).

The entry or element in the ith row and jth column is a real number and is denoted by the double-subscript notation aij. We call aij the (i, j) th entry. For example, a24 is the entry in the second row and fourth column of the matrix A. In general, we will use capital letters A, B, … to denote matrices and the corresponding lowercase letter aij,bij, … for their entries.

If A has n rows and n columns, then A is called a square matrix of order n. The entries a11,  a22, …,  ann form the main diagonal of A. A 1×n matrix is called a row matrix, and an n×1 matrix is called a column matrix.

Example 1 Determining the Order of Matrices

Determine the order of each matrix. Identify square, row, and column matrices. Identify entries in the main diagonal of each square matrix.

  1. A=[3]

  2. B=[35−7]

  3. C=[01−34]

  4. D=[123456789]

Solution

  1. Matrix A with one row and one column is a 1×1 matrix. A is a square matrix of order 1. In A, a11=3 is the main diagonal. A is also a column and a row matrix.

  2. Matrix B with one row and three columns is a 1×3 matrix. B is a row matrix.

  3. Matrix C, a 2×2 matrix, is a square matrix of order 2. In C, the entries c11=0 and c22=4 form the main diagonal.

  4. Matrix D, a 3×3 matrix, is a square matrix of order 3. In D, the entries d11=1, d22=5, and d33=9 form the main diagonal.

Practice Problem 1

  1. Determine the order of each matrix.

    1. [−137400]

    2. [3−8]

Using Matrices to Solve Linear Systems

  1. 2 Use matrices to solve a system of linear equations.

To solve a system of linear equations by the elimination method, the particular symbols used for the variables do not matter; only the coefficients and the constants are important. We can display the constants and coefficients of a system in a matrix called the augmented matrix of the system. The matrix containing only the coefficients of the variables is the coefficient matrix. Consider the following system.

{x−y−z=12x−3y+z=10x+y−2z=0

Augmented Matrix:

Coefficient Matrix:

 [1−1−12−3111−2]

Notice that the numbers in the first column of the augmented matrix are the coefficients of x, those in the second column are the coefficients of y, and those in the third column are the coefficients of z. The constants on the right side of the equations in the system are found in the fourth column. The vertical line in the augmented matrix is to remind you of the equal signs in the equations. If a variable does not appear in one of the equations, represent it with a zero in the matrix.

Example 2 Writing the Augmented Matrix of a Linear System

Write the augmented matrix of the linear system.

{2x+3z=12z+y=5−4x+5y=7

Solution

First, write the system with the variables lined in columns and insert zeros as coefficients of any missing variables.

{2x+0y+3z=10x+1y+2z=5−4x+5y+0z=7

The augmented matrix of the given system is:

[20310125−4507].

Practice Problem 2

  1. Write the augmented matrix of the linear system.

    {3y−z=08x+4y=14−2y+9z=0

We can reverse this process and write a linear system from a given augmented matrix. For example, the augmented matrix

[92−16347−801511] corresponds to the system of equations {9x+2y−z=6 3x+4y+7z=−8.y+5z=11

The basic strategy for solving a system of equations is to replace the given system with an equivalent system (one with the same solution set) that is easier to solve. In the previous chapter, we used three basic operations to solve a linear system:

  1. Interchange two equations.

  2. Multiply all the terms in an equation by a nonzero constant.

  3. Add a multiple of one equation to another equation. In other words, replace one equation with the sum of itself and a multiple of another equation.

In matrix terminology, the three corresponding operations are called the elementary row operations. Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations, given next. The symbol Ri represents the ith row of a matrix.

To add two rows, we just add the numbers in the corresponding entries of each row.

To multiply the row by a number c, we just multiply each entry of the row by c.

Example 3 Applying Elementary Row Operations

Perform the indicated row operations (a), (b), and (c) in order on the following matrix:

A=[3−2−32414].

  1. R1↔R2,

  2. 12R1,

  3. −3R1+R2→R2

Solution

  1. A=[3−2−32414]R1↔R2→[24143−2−3]=BInterchange the 1stand 2nd rows.

  2. B=[24143−2−3]12R1→[1273−2−3]=CMultiply the 1st row by12.

  3. C=[1273−2−3]−3R1+R2→R2→[1270−8−24]Add −3 times the 1strow to the 2nd row.

Practice Problem 3

  1. Perform the row operations of Example 3 in order on the following matrix:

    A=[345246]

In the next example, we solve a system of equations both with and without matrix notation and place the results side by side for comparison. We solve the system by converting it into triangular form by first using elimination and then back-substitution.

Example 4 Comparing Linear Systems and Matrices

Solve the system of linear equations:

{x−y−z=1(1)2x−3y+z=10(2)x+y−2z=0(3)

Solution

Linear System Augmented Matrix
{x−y−z=1(1)2x−3y+z=10(2)x+y−2z=0(3) A=[1−1−112−311011−20]
{x−y−z=1(1)−y+3z=8(4)2y−z=−1(5) −2R1+R2→R2→(−1)R1+R3→R3→[1−1−110−13802−1−1]
{x−y−z=1(1)−y−3z=−8(6)2y−z=−1(5) (−1)R2→[1−11101−3−802−1−1]
{x−y−z=1(1)y−3z=−8(6)5z=15(7) −2R2+R3→R3→[1−1−1101−3−800515]

Equation (7), 5z=15, which corresponds to row 3 of the final matrix, gives the value z=3. We find y by back-substitution.

y−3z=−8Equation (6); final matrix row 2y−3(3)=−8Replace z with 3.y=1Solve for y.

We find x by back-substitution.

x−y−z=1Equation (1); final matrix row 1x−1−3=1Replace y with 1 and z with 3.x=5Solve for x.

The solution of the system is x=5, y=1, and z=3. You should check this solution by substituting it into the original system of equations. The solution set is {(5, 1, 3)}.

Practice Problem 4

  1. Solve the system of linear equations:

    {x−6y+3z=−23x+3y−2z=−22x−3y+z=−2

The last matrix in the solution of Example 4 is in row-echelon form, which is defined next. In the definition, a nonzero row means a row that contains at least one nonzero entry; a leading entry of a row is the leftmost nonzero entry in a nonzero row.

Property 2 says that the leading entries form an echelon (steplike) pattern that moves down and to the right.

The following matrices are in row-echelon form; a starred entry may be any value, including zero.

The following matrices are in reduced row-echelon form because the entries above and below each leading 1 are zero. (Again, the starred entries may be any value, including zero.)

[010000100001][100*010*001*][1*00001000010000]

An augmented matrix may be transformed into several different row-echelon form matrices by using different sequences of row operations. However, its reduced row-echelon form is unique. See Exercise 97.

Gaussian Elimination

  1. 3 Use Gaussian elimination to solve a system.

The method of solving a system of linear equations by transforming the augmented matrix of the system into row-echelon form and then using back-substitution to find the solution set is known as Gaussian elimination. See Section 5.2.

Practice Problem 5

  1. Solve by Gaussian elimination.

    {x−2y=12x+3y=16

Side Note

Note that when we perform a row operation such as

3R1+R2→R2 on a matrix,

  1. the entries in row 1 are unchanged and

  2. we multiply each entry of row 1 by 3 and add this product to the corresponding entries of row 2 to obtain a new row 2.

Example 6 Solving a System by Using Gaussian Elimination

Solve by Gaussian elimination.

{2x+y+z=6−3x−4y+2z=4x+y−z=−2

Solution

  1. Step 1 A=[2116−3−42411−1−2]The augmented matrixof the system

  2. Step 2 R1↔R3→[11−1−2−3−4242116]3R1+R2→R2→−2R1+R3→R3→[11−1−20−1−1−20−1310](−1)R2→[11−1−201120−1310]R2+R3→R3→[11−1−2011200412]14R3→[11−1−201120013]

    The last matrix is in row-echelon form.

  3. Step 3 The system of equations corresponding to the last matrix in Step 2 is

    {x+y−z=−2(1)y+z=2(2)z=3(3)

  4. Step 4 Equation (3) in Step 3 gives the value z=3. Back-substitute z=3 in equation (2).

    y+z=2Equation (2)y+3=2Replace z with 3.y=−1Solve for y.

    Now back-substitute z=3 and y=−1 in equation (1).

    x+y−z=−2Equation (1)x−1−3=−2x=2Solve for x.

    You should check the solution set, {(2, −1, 3)}, by substituting these values into the original system of equations.

Practice Problem 6

  1. Solve the system of equations:

    {2x+y−z=7x−3y−3z=44x+y+z=3

Example 7 Attempting to Solve a System with No Solution

Solve the system of equations by Gaussian elimination:

{y+5z=−4x+4y+3z=−22x+7y+z=8

Solution

  1. Step 1 A=[015−4143−22718]The augmented matrix of the system

  2. Step 2 R1↔R2→[143−2015−42718]

    −2R1+R3→R3→[143−2015−40−1−512]Row 2 already begins with a 0.R2+R3→R3→[143−2015−40008]18R3→[143−2015−40001]The matrix is now in row-echelonform.

  3. Step 3 The system of equations corresponding to the last matrix in Step 2 is

    {x+4y+3z=−2y+5z=−40=1A false statement

    Because the third equation 0=1 is a false statement, we conclude that this system is inconsistent. Because this system is equivalent to the original system, the original system is also inconsistent.

  4. Step 4 The solution set for the system is ∅.

Practice Problem 7

  1. Solve the following system of equations by first transforming the augmented matrix into row-echelon form.

    {6x+8y−14z=33x+4y−7z=126x+3y+z=0

Gauss–Jordan Elimination

  1. 4 Use Gauss–Jordan elimination to solve a system.

If we continue the Gaussian elimination procedure until we have a reduced row-echelon form, the procedure is called Gauss–Jordan elimination.

Example 8 Solving a System of Equations by Gauss–Jordan Elimination

Solve the system given in Example 4 by Gauss–Jordan elimination.

{x−y−z=12x−3y+z=10The given systemx+y−2z=0

Solution

[1−1−1101−3−800215]The final augmented matrix of the systemin Example 415R3→[1−1−1101−3−80013]The matrix is now in row-echelon form.R2+R1→R1→[10−4−701−3−80013]4R3+R1→R1→3R3+R2→R2→[100501010013]

We now have an equivalent matrix in reduced row-echelon form. The corresponding system of equations for the last augmented matrix is as follows:

{x=5y=1z=3

The solution set is therefore {(5, 1, 3)}, as in Example 4.

Practice Problem 8

  1. Solve the system by Gauss–Jordan elimination:

    {2x−3y−2z=0x+y−2z=73x−5y−5z=3

Example 9 Solving a System with Infinitely Many Solutions

Solve the system of equations:

{x+2y+5z=4y+4z=42x+4y+10z=8

Solution

The augmented matrix A of the system is given below. We want to find the equivalent system in reduced row-echelon form.

Because a11=1 and a21=0, we need a zero at the (3, 1) position.

A=[1254014424108]−2R1+R3→R3→[125401440000]

Next, we need a zero at the (1, 2) position.

−2R2+R1→R1→[10−3−401440000]The matrix is now in reducedrow-echelon form.

The equivalent system is

x−3z=−4y+4z=4

Solving for x and y in terms of z, we obtain

x=3z−4y=−4z+4

Each real number z results in a solution with y=−4z+4 and x=3z−4, giving infinitely many solutions of the form (3z−4, −4z+4, z). The solution set is {(3z−4, −4z+4, z)}.

Practice Problem 9

  1. Solve the system of equations:

    {x+z=−13y+2z=53x−3y+z=−8

Example 10 Leontief Input–Output Model

Consider an economy that has steel, coal, and transportation industries. There are two types of demands (measured in dollars) on the production of each industry: interindustry demand and external consumer demand. The outputs and requirements of the three industries are shown in Figure 6.1. For example, $1.00 of transportation output requires $0.10 from steel and $0.01 from coal.

Figure 6.1

Output diagram of an economy

  1. Write a system of equations that expresses the outputs of the three industries. Assume that all quantities are given in millions of dollars.

  2. Verify that s=15, c=10, and t=8 will meet both interindustry and consumer demand.

Solution

  1. To satisfy both consumer and interindustry demand, we obtain the following system of outputs. (s denotes total steel output; c, total coal output; t, total transportation output.)

    {s=0.01s+0.1t+0.1c+13.05Distribution of steel outputc=0.2s+0.01t+0.02c+6.72Distribution of coal outputt=0.25s+0.3c+1.25Distribution of transportation output

    You can rewrite this system as follows:

    {0.99s−0.1c−0.1t=13.05(1)−0.2s+0.98c−0.01t=6.72(2)−0.25s−0.3c+t=1.25(3)

  2. To verify that the given numbers (obtained by solving the system above using matrices) satisfy these equations, we substitute s=15, c=10, and t=8 into each equation.

    (0.99)(15)−(0.1)(10)−(0.1)(8)=13.05✓(1)(−0.20)(15)+(0.98)(10)−(0.01)(8)=6.72✓(2)(−0.25)(15)−(0.30)(10)+8=1.25✓(3)

    In other words, output levels of s=15, c=10, and t=8 million dollars will meet interindustry and consumer demand.

Practice Problem 10

  1. In Example 10 , assume that consumer demand (in millions of dollars) is 12.46 for steel, 3 for coal, and 2.7 for transportation.

    1. Write a system of equations to express the outputs for the three industries.

    2. Verify that s=14, c=6, t=8 will meet both interindustry and consumer demand.

Section 6.1 Exercises

Concepts and Vocabulary

  1. A matrix is any rectangular array of                     .

  2. The array of coefficients and constants in a linear system is called the                           of the system.

  3. Two matrices are row-equivalent if one can be obtained from the other by a sequence of                          .

  4. If a matrix is in row-echelon form and each leading entry 1 is the only nonzero entry in its column, then the matrix is in                           form.

  5. True or False. If A is a 3×4 matrix, then each row of A has three entries.

  6. True or False. Every m×n (n≥2) matrix is the matrix of some linear system.

  7. True or False. The augmented matrix for the system of equations

    {2x+3y=53y−4z=−1x+2z=3is[2353−4−1123].

  8. True or False. The augmented matrix for the system of equations

    {2x−y=13x+y=95x−2y=4is[2x−y13xy95x−2y4].

Building Skills

In Exercises 9–14, determine the order of each matrix.

  1. [7]

  2. [1359]

  3. [34.37.582−1−30]

  4. [−1−234−571−00]

  5. [451000]

  6. [−52−372.5e−π12−eπ−01]

  7. Let A=[123456789101112]. Identify the entries a13,  a31,  a33, and a34.

  8. In matrix A from Exercise 15, identify the (i, j)th location aij of each entry.

    1. 7

    2. 10

    3. 4

    4. 12

  9. Is the following array a matrix? Why or why not?

  10. Write the matrix A with the following entries:

    a13=5,  a22=2,  a11=6,  a12=−5,  a23=4,  and a21=7

In Exercises 19–24, write the augmented matrix for each system of linear equations.

  1. {2x+4y=2x−3y=1

  2. {x+2y=73x+5y=11

  3. {5x−11=717y−19=13x

  4. {2y−10=3x5x+7=y

  5. {−x+2y+3z=82x−3y+9z=164x−5y−6z=32

  6. {x−y=22x+3z=−5y−2z=7

In Exercises 25–28, write the system of linear equations represented by each augmented matrix. Use x, y, and z as the variables.

  1. [12−34−2−3153−327]

  2. [−123623124321]

  3. [1−11221−36]

  4. [11121−1−142316−11−18]

In Exercises 29–32, perform the indicated elementary row operations in the stated order.

  1. [235123];

    1. R1↔R2,

    2. −2R1+R2→R2,

    3. −R2

  2. [242157];

    1. 12R1,

    2. (−1)R1+R2→R2,

    3. 13R2

  3. [123404−311015−3];

    1. R2↔R3,

    2. −4R2+R3→R3,

    3. −123R3

  4. [132−2120234−2−173];

    1. 2R1+R3→R3,

    2. 12R2+R3→R3

In Exercises 33–36, identify the elementary row operation used and supply the missing entries in each row-equivalent matrix.

  1. [45−754−2]→[1?−74542]→[1?−740−94?]→[1?−7401?]

  2. [26−83−12]→[13−?3−12]→[13?0−?14]→[13−?01?]

  3. [14310−3−20075−3]→[143101?0075−3]→[143101?0001?]→[143101?0001?]

  4. [111323381−3−25]→[111301?21−3−25]→[111301?20−4?2]→[111301?200?10]

In Exercises 37–44, determine whether each matrix is in row-echelon form. If your answer is no, explain why. If your answer is yes, is the matrix in reduced row-echelon form?

  1. [012103]

  2. [125036]

  3. [100201030014]

  4. [001201011003]

  5. [12020015]

  6. [01−1−20200000120]

  7. [100−2010300120000]

  8. [1003010400050013]

In Exercises 45–54, the augmented matrix of a system of equations has been transformed to an equivalent matrix in row-echelon form or reduced row-echelon form. Using x, y, z, and w as variables, write the system of equations corresponding to the matrix. If the system is consistent, solve it.

  1. [12101−2]

  2. [102013]

  3. [14220013]

  4. [12340001]

  5. [123201−24001−1]

  6. [10212013120015]

  7. [100020100−50012300000]

  8. [100040100300012]

  9. [1000−5010040012300010]

  10. [10003010020010000001]

In Exercises 55–68, solve each system of equations by Gaussian elimination.

  1. {x−2y=112x−y=13

  2. {3x−2y=44x−3y=5

  3. {2x−3y=34x−y=11

  4. {3x+2y=16x+4y=3

  5. {3x−5y=44x−15y=13

  6. {−2x+4y=13x−5y=−9

  7. {x−y=12x+y=53x−4y=2

  8. {y=2x+13x+2y+1.5=04x−2y+2=0

  9. {x+y+z=6x−y+z=22x+y−z=1

  10. {2x+4y+z=5x+y+z=62x+3y+z=6

  11. {2x+3y−z=9x+y+z=93x−y−z=−1

  12. {x+y+2z=42x−y+3z=93x−y−z=2

  13. {3x+2y+4z=192x−y+z=36x+7y−z=17

  14. {4x+3y+z=82x+y+4z=−43x+z=1

In Exercises 69–74, solve each system of equations by Gauss–Jordan elimination.

  1. {x−y=1x−z=−12x+y−z=3

  2. {4x+5z=7y−6z=83x+4y=9

  3. {x+y−z=4x+3y+5z=103x+5y+3z=18

  4. {x+y+z=−52x−y−z=−4y+z=−2

  5. {x+2y−z=63x+y+2z=32x+5y+3z=9

  6. {2x+4y−z=9x+3y−3z=43x+y+2z=7

In Exercises 75 and 76, the reduced row-echelon forms of the augmented matrices of three systems of equations are given. How many solutions does each system have?

    1. [102013]

    2. [123800120000]

    3. [102003]

    1. [100−101020013]

    2. [100401250006]

    3. [112300000000]

In Exercises 77 and 78, determine whether the statement is true or false and justify your answer.

  1. If A is a 5×7 matrix, then each column of A has seven entries.

  2. If a matrix A is in reduced row-echelon form, then at least one of the entries in each column must be a 1.

Applying the Concepts

In Exercises 79–82, Leontief’s input–output model of a simplified economy is described. See Example 10. For each exercise, do the following.

  1. Set up (without solving) a linear system whose solution will represent the required production schedule.

  2. Write an augmented matrix for the economy.

  3. Solve the system of equations by using part (b) to find the production schedule that will meet interindustry and consumer demand.

  1. Two-sector economy. Consider an economy with only two industries: A and B. Suppose industry B needs $0.10 worth of A’s product for each $1.00 of output B produces and that industry A needs $0.20 worth of B’s product for each $1.00 of output A produces. Consumer demand for A’s product is $1000, and consumer demand for B’s product is $780.

  2. A company with two branches. A company has two interacting branches: A and B. Branch A consumes $0.50 of its own output and $0.20 of B’s output for every $1.00 it produces. Branch B consumes $0.60 of A’s output and $0.30 of its own output for $1.00 of output. Consumer demand for A’s product is $50,000, and consumer demand for B’s product is $40,000.

  3. Three-sector economy. An economy has three sectors: labor, transportation, and food industries. Suppose the demand on $1.00 in labor is $0.40 for transportation and $0.20 for food; the demand on $1.00 in transportation is $0.50 for labor and $0.30 for transportation; and the demand on $1.00 in food production is $0.50 for labor, $0.05 for transportation, and $0.35 for food. Outside consumer demand for the current production period is $10,000 for labor, $20,000 for transportation, and $10,000 for food.

  4. Three-sector economy. Consider an economy with three industries A, B, and C, with outputs a, b, and c, respectively. Demand on the three industries is shown in the figure.

  5. Heat transfer. In a study of heat transfer in a grid of wires, the temperature at an exterior node is maintained at a constant value (in °F) as shown in the accompanying figure. When the grid is in thermal equilibrium, the temperature at an interior node is the average of the temperatures at the four adjacent nodes. For instance, T1=0+0+300+T24, or 4T1−T2=300. Find the temperatures T1, T2, and T3 when the grid is in thermal equilibrium.

  6. Heat transfer. Repeat Exercise 83 for the following grid of wires.

  7. Traffic flow. The sketch shows the intersections of certain one-way streets. The traffic volume during one hour was observed. To keep the traffic moving, traffic controllers use the formula that the number of cars per hour entering an intersection equals the number of cars per hour exiting the intersection.

    1. Set up a system of equations that keeps traffic moving.

    2. Solve the system of equations in part (a).

  8. Traffic flow. Repeat Exercise 85 for the following pattern of one-way streets.

  9. Parabola. Find the equation of the parabola y=ax2+bx+c that passes through the points (−1, 9), (1, 3), and (2, 6).

  10. Modeling. A football was punted. Its height h above the ground at time t is given by the following table:

    Time t (seconds) Height h (feet)
    t=0.5 31
    t=1 51
    t=2 67

    1. Use the system of equations to find the equation of the parabola of the form h=at2+bt+c.

    2. What is the hang time (the time it takes the ball to touch the ground) of the punt?

    3. What is the maximum height of the ball?

  11. Curve fitting. Find the function f(x)=a+b sin(x)+c sin(2x) c that passes through the points (π4,−3), (3π4,2), and (7π4,5).

  12. Curve fitting. Find the function f(x)=a+b cos(x)+c cos(2x) that passes through the points (π6,2), (π3,1), and (4π6,3).

  13. Find the cubic function y=ax3+bx2+cx+d whose graph passes through the points (1, 5), (−1, 1),(2, 7), and (−2, 11).

  14. Find the cubic function y=ax3+bx2+cx+d whose graph passes through the points (1, 8), (−1, 2), (2, 8), and (−2, 20).

  15. Curve fitting. Find the piecewise linear function f(x)={ax+bcx+d that produces the following graph.

  16. Curve fitting. Find the piecewise linear function f(x)={ax+bcx+d that produces the following graph.

Beyond the Basics

In Exercises 95 and 96, solve each system of equations by Gauss–Jordan elimination.

  1. {x+y+z+w=0x+3y+2z+4w=02x+z−w=0

  2. {x−y+z−w=4x+2y+z+w=22x+3y+4z+5w=53x+4y+2z−w=8

  3. Let A=[12−3110−320112230−2]

    1. Find two different matrices B and C in row-echelon form that are row-equivalent to A.

    2. Show that the reduced row-echelon form of the matrices B and C of part (a) produce the same matrix.

  4. Consider the following system of linear equations:

    {x+y+z=6x−y+z=22x+y+z=1

    1. Find two different matrices B and C in row-echelon form that yield the same solution set.

    2. Show that the reduced row-echelon form of the matrices B and C of part (a) produces the same matrix.

    1. Use the methods of this section to solve a general 2×2 system:

      {ax+by=mcx+dy=n

    2. Under what conditions on the coefficients does the system have (i) a unique solution, (ii) no solution, and (iii) infinitely many solutions?

  6. Construct three different augmented matrices for linear systems whose solution set is x=1, y=0, z=−2. There are many different correct answers.

In Exercises 101 and 102, solve each system of equations by using row operations.

  1. {log+log y+log z=63 log−log y+3 logz=105 log+5 logy−4 logz=3

    [Hint: Let u=log x, v=log y, and w=log z.]

  2. {4.2x−3.3y+5z=12x+4.3y−2.5z=102.2x−2.3y+3.5z=4

    [Hint: Let u=2x, v=3y, and w=5z.]

Collinear points. Three points P(x1, y1), Q(x2, y2), and R(x3, y3) are collinear if they lie on the same line—that is, if all three points P, Q, and R satisfy the equation Ax+By=C for some A, B, and C, (where at least one of A, B or C is not zero). In terms of systems of linear equations, P, Q, and R are collinear if the corresponding system of linear equations

{Ax1+By1=CAx2+By2=CAx3+By3=C

has a nontrivial solution (a solution where at least one of A, B, or C is not zero).

In Exercises 103 and 104, use the Gaussian elimination method to determine if the given points are collinear.

  1. (2, 1), (1/3, −1/9), and (−3, −7/3)

  2. (−1, 3/2), (1, 5/2), and (2, 4)

Coplanar points. Four points P(x1, y1, z1), Q(x2, y2, z2), R(x3, y3, z3), and S(x4, y4, z4) are coplanar if they lie on the same plane—that is, if all four points P, Q, R, and S satisfy the equation Ax+By+Cz=D for some A, B, C and D (where at least one of A, B, C, or D is not zero). In terms of systems of linear equations, P, Q, R, and S are coplanar if the corresponding system of linear equations

{Ax1+By1+Cz1=DAx2+By2+Cz2=DAx3+By3+Cz3=DAx4+By4+Cz4=D

has a nontrivial solution (a solution where at least one of A, B, C, or D is not zero).

In Exercises 105 and 106, use the Gaussian elimination method to determine if the given points are coplanar.

  1. (2,1,−5/4),(−1,0−3/2),(4,3,−7/4) and (1,−2,0)

  2. (1, 1, 5),(−2, 1, 4), (1, −3, −10) and (4, 0, 10)

  3. The general theory of finite differences tells us that

    1+4+7+⋯+(3n−2)=P(n)

    where P(n) is a quadratic function of n. Find P(n).

  4. The general theory of finite differences tells us that

    12+32+52+⋯+(2n−1)2=Q(n)

    where Q(n) is a cubic polynomial of n. Find Q(n).

Critical Thinking/Discussion/Writing

  1. Find the form of all 2×1 matrices in reduced row-echelon form.

  2. Find the form of all 2×2 matrices in reduced row-echelon form.

  3. Suppose a matrix A is transformed to a matrix B by an elementary row operation. Is there an elementary row operation that transforms B to A? Explain.

  4. Are the following statements true or false? Justify your answers. Suppose a matrix A is in reduced row-echelon form.

    1. If we delete a row of A, then the remaining matrix is in reduced row-echelon form.

    2. Repeat part (a), replacing row with column.

Getting Ready for the Next Section

In Exercises 113–116, solve each equation.

  1. 3(x−1)=5−x

  2. 5(x−2)=3(x−3)+13

  3. −3(x+4)+2=8−x

  4. 6x−3(5x+2)=4−5x

In Exercises 117–120, solve each system.

  1. {2x−y=5x+2y=25

  2. {x−3y=12x+y=−5

  3. {x+3y=62x+6y=8

  4. {2x−y=36x−3y=9

In Exercises 121–124, identify each statement as True or False.

  1. If x(3t2+π√t4+t2+5)=0 where t is a real number, then x=0.

  2. If x, y, and z are any real numbers, then xy+z=x+yz.

  3. If x, y, and z are real numbers, then xy(z+5)=xyz+5xy.

  4. If (x4+x2)(y−3)=1 and y=103, then x4+x2=3.

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