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2 Chapter 1 Systems of Linear Equations and Matrices 1.1 Introduction to Systems of Linear Equations Systems of linear equations and their solutions constitute one of the major topics that we will study in this course. In this ﬁrst section we will introduce some basic terminology and discuss a method for solving such systems. Linear Equations Recall that in two dimensions a line in a rectangular xy -coordinate system can be repre- sented by an equation of the form ax + by = c (a, b not both 0) and in three dimensions a plane in a rectangular xyz-coordinate system can be repre- sented by an equation of the form ax + by + cz = d (a, b, c not all 0) These are examples of “linear equations,” the ﬁrst being a linear equation in the variables x and y and the second a linear equation in the variables x , y , and z. More generally, we deﬁne a linear equation in the n variables x1 , x2 ,... , xn to be one that can be expressed in the form a1 x1 + a2 x2 + · · · + an xn = b (1) where a1 , a2 ,... , an and b are constants, and the a ’s are not all zero. In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equations as a1 x + a2 y = b (a1 , a2 not both 0) (2) a1 x + a2 y + a3 z = b (a1 , a2 , a3 not all 0) (3) In the special case where b = 0, Equation (1) has the form a1 x1 + a2 x2 + · · · + an xn = 0 (4) which is called a homogeneous linear equation in the variables x1 , x2 ,... , xn. E X A M P L E 1 Linear Equations Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the ﬁrst power and do not appear, for example, as arguments of trigonometric, logarithmic, or exponential functions. The following are linear equations: x + 3y = 7 x1 − 2x2 − 3x3 + x4 = 0 1 2 x − y + 3z = −1 x1 + x2 + · · · + xn = 1 The following are not linear equations: x + 3y 2 = 4 3x + 2y − xy = 5 √ sin x + y = 0 x1 + 2x2 + x3 = 1 A ﬁnite set of linear equations is called a system of linear equations or, more brieﬂy, a linear system. The variables are called unknowns. For example, system (5) that follows has unknowns x and y , and system (6) has unknowns x1 , x2 , and x3. 5x + y = 3 4x1 − x2 + 3x3 = −1 (5–6) 2x − y = 4 3x1 + x2 + 9x3 = −4 1.1 Introduction to Systems of Linear Equations 3 A general linear system of m equations in the n unknowns x1 , x2 ,... , xn can be written The double subscripting on as the coefﬁcients aij of the un- a11 x1 + a12 x2 + · · · + a1n xn = b1 knowns gives their location in the system—the ﬁrst sub- a21 x1 + a22 x2 + · · · + a2n xn = b2........ (7) script indicates the equation.... in which the coefﬁcient occurs, am1 x1 + am2 x2 + · · · + amn xn = bm and the second indicates which A solution of a linear system in n unknowns x1 , x2 ,... , xn is a sequence of n numbers unknown it multiplies. Thus, a12 is in the ﬁrst equation and s1 , s2 ,... , sn for which the substitution multiplies x2. x1 = s1 , x2 = s2 ,... , xn = sn makes each equation a true statement. For example, the system in (5) has the solution x = 1, y = − 2 and the system in (6) has the solution x1 = 1, x2 = 2, x3 = −1 These solutions can be written more succinctly as (1, −2) and (1, 2, −1) in which the names of the variables are omitted. This notation allows us to interpret these solutions geometrically as points in two-dimensional and three-dimensional space. More generally, a solution x1 = s1 , x2 = s2 ,... , xn = sn of a linear system in n unknowns can be written as (s1 , s2 ,... , sn ) which is called an ordered n-tuple. With this notation it is understood that all variables appear in the same order in each equation. If n = 2, then the n-tuple is called an ordered pair, and if n = 3, then it is called an ordered triple. Linear Systems inTwo and Linear systems in two unknowns arise in connection with intersections of lines. For Three Unknowns example, consider the linear system a1 x + b1 y = c1 a2 x + b2 y = c2 in which the graphs of the equations are lines in the xy-plane. Each solution (x, y) of this system corresponds to a point of intersection of the lines, so there are three possibilities (Figure 1.1.1): 1. The lines may be parallel and distinct, in which case there is no intersection and consequently no solution. 2. The lines may intersect at only one point, in which case the system has exactly one solution. 3. The lines may coincide, in which case there are inﬁnitely many points of intersection (the points on the common line) and consequently inﬁnitely many solutions. In general, we say that a linear system is consistent if it has at least one solution and inconsistent if it has no solutions. Thus, a consistent linear systemof two equations in 4 Chapter 1 Systems of Linear Equations and Matrices y y y x x x No solution One solution Infinitely many solutions (coincident lines) Figure 1.1.1 two unknowns has either one solution or inﬁnitely many solutions—there are no other possibilities. The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again we see that there are only three possibilities—no solutions, one solution, or inﬁnitely many solutions (Figure 1.1.2). No solutions No solutions No solutions No solutions (three parallel planes; (two parallel planes; (no common intersection) (two coincident planes no common intersection) no common intersection) parallel to the third; no common intersection) One solution Infinitely many solutions Infinitely many solutions Infinitely many solutions (intersection is a point) (intersection is a line) (planes are all coincident; (two coincident planes; intersection is a plane) intersection is a line) Figure 1.1.2 We will prove later that our observations about the number of solutions of linear systems of two equations in two unknowns and linear systems of three equations in three unknowns actually hold for all linear systems. That is: Every system of linear equations has zero, one, or inﬁnitely many solutions. There are no other possibilities. 1.1 Introduction to Systems of Linear Equations 5 E X A M P L E 2 A Linear System with One Solution Solve the linear system x−y =1 2x + y = 6 Solution We can eliminate x from the second equation by adding −2 times the ﬁrst equation to the second. This yields the simpliﬁed system x−y =1 3y = 4 From the second equation we obtain y = 43 , and on substituting this value in the ﬁrst equation we obtain x = 1 + y = 73. Thus, the system has the unique solution x = 73 , y = 4 3 Geometrically, this means that 7 the lines represented by the equations in the system 4 intersect at the single point 3 , 3. We leave it for you to check this by graphing the lines. E X A M P L E 3 A Linear System with No Solutions Solve the linear system x+ y=4 3 x + 3y = 6 Solution We can eliminate x from the second equation by adding −3 times the ﬁrst equation to the second equation. This yields the simpliﬁed system x+y = 4 0 = −6 The second equation is contradictory, so the given system has no solution. Geometrically, this means that the lines corresponding to the equations in the original system are parallel and distinct. We leave it for you to check this by graphing the lines or by showing that they have the same slope but different y -intercepts. E X A M P L E 4 A Linear System with Inﬁnitely Many Solutions Solve the linear system 4 x − 2y = 1 16x − 8y = 4 Solution We can eliminate x from the second equation by adding −4 times the ﬁrst equation to the second. This yields the simpliﬁed system 4 x − 2y = 1 0=0 The second equation does not impose any restrictions on x and y and hence can be omitted. Thus, the solutions of the system are those values of x and y that satisfy the single equation 4x − 2y = 1 (8) Geometrically, this means the lines corresponding to the two equations in the original system coincide. One way to describe the solution set is to solve this equation for x in terms of y to obtain x = 41 + 21 y and then assign an arbitrary value t (called a parameter) 6 Chapter 1 Systems of Linear Equations and Matrices to y. This allows us to express the solution by the pair of equations (called parametric In Example 4 we could have equations) also obtained parametric equations for the solutions x= 1 4 + 21 t, y = t by solving (8) for y in terms We can obtain speciﬁc numerical solutions from these equations by substituting 1 numer- of x and letting x = t be ical values for the parameter t. For example, t = 0 yields the solution 4 ,0 , t = 1 the parameter. The resulting yields the solution 43 , 1 , and t = −1 yields the solution − 41 , −1. You can conﬁrm parametric equations would that these are solutions by substituting their coordinates into the given equations. look different but would deﬁne the same solution set. E X A M P L E 5 A Linear System with Inﬁnitely Many Solutions Solve the linear system x − y + 2z = 5 2x − 2y + 4z = 10 3x − 3y + 6z = 15 Solution This system can be solved by inspection, since the second and third equations are multiples of the ﬁrst. Geometrically, this means that the three planes coincide and that those values of x , y , and z that satisfy the equation x − y + 2z = 5 (9) automatically satisfy all three equations. Thus, it sufﬁces to ﬁnd the solutions of (9). We can do this by ﬁrst solving this equation for x in terms of y and z, then assigning arbitrary values r and s (parameters) to these two variables, and then expressing the solution by the three parametric equations x = 5 + r − 2s, y = r, z = s Speciﬁc solutions can be obtained by choosing numerical values for the parameters r and s. For example, taking r = 1 and s = 0 yields the solution (6, 1, 0). Augmented Matrices and As the number of equations and unknowns in a linear system increases, so does the Elementary Row Operations complexity of the algebra involved in ﬁnding solutions. The required computations can be made more manageable by simplifying notation and standardizing procedures. For example, by mentally keeping track of the location of the +’s, the x ’s, and the =’s in the linear system a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2............ am1 x1 + am2 x2 + · · · + amn xn = bm we can abbreviate the system by writing only the rectangular array of numbers ⎡ ⎤ a11 a12 · · · a1n b1 ⎢ ⎥ ⎢a21 a22 · · · a2 n b2 ⎥ ⎢....... ⎥ As noted in the introduction ⎣..... ⎦ to this chapter, the term “ma- am1 am2 · · · amn bm trix” is used in mathematics to denote a rectangular array of This is called the augmented matrix for the system. For example, the augmented matrix numbers. In a later section for the system of equations we will study matrices in de- ⎡ ⎤ tail, but for now we will only x1 + x2 + 2x3 = 9 1 1 2 9 ⎢ ⎥ be concerned with augmented 2 x 1 + 4 x 2 − 3x 3 = 1 is ⎣2 4 −3 1⎦ matrices for linear systems. 3x1 + 6x2 − 5x3 = 0 3 6 −5 0 1.1 Introduction to Systems of Linear Equations 7 The basic method for solving a linear system is to perform algebraic operations on the system that do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: 1. Multiply an equation through by a nonzero constant. 2. Interchange two equations. 3. Add a constant times one equation to another. Since the rows (horizontal lines) of an augmented matrix correspond to the equations in the associated system, these three operations correspond to the following operations on the rows of the augmented matrix: 1. Multiply a row through by a nonzero constant. 2. Interchange two rows. 3. Add a constant times one row to another. These are called elementary row operations on a matrix. In the following example we will illustrate how to use elementary row operations and an augmented matrix to solve a linear system in three unknowns. Since a systematic procedure for solving linear systems will be developed in the next section, do not worry about how the steps in the example were chosen. Your objective here should be simply to understand the computations. E X A M P L E 6 Using Elementary Row Operations In the left column we solve a system of linear equations by operating on the equations in the system, and in the right column we solve the same system by operating on the rows of the augmented matrix. ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ 2 x + 4 y − 3z = 1 ⎣2 4 −3 1⎦ 3 x + 6y − 5 z = 0 3 6 −5 0 Add −2 times the ﬁrst equation to the second Add −2 times the ﬁrst row to the second to to obtain obtain ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ 2y − 7z = −17 ⎣0 2 −7 −17⎦ 3x + 6y − 5z = 0 3 6 −5 0 Historical Note The ﬁrst known use of augmented matrices appeared between 200 B.C. and 100 B.C. in a Chinese manuscript entitled Nine Chapters of Mathematical Art. The coefﬁcients were arranged in columns rather than in rows, as today, but remarkably the system was solved by performing a succession of operations on the columns. The actual use of the term augmented matrix appears to have been intro- duced by the American mathematician Maxime Bôcher in his book In- troduction to Higher Algebra, published in 1907. In addition to being an outstanding research mathematician and an expert in Latin, chemistry, philosophy, zoology, geography, meteorology, art, and music, Bôcher was an outstanding expositor of mathematics whose elementary text- books were greatly appreciated by students and are still in demand Maxime Bôcher today. (1867–1918) [Image: Courtesy of the American Mathematical Society www.ams.org] 8 Chapter 1 Systems of Linear Equations and Matrices Add −3 times the ﬁrst equation to the third to Add −3 times the ﬁrst row to the third to obtain obtain ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ 2y − 7z = −17 ⎣0 2 −7 −17⎦ 3y − 11z = −27 0 3 −11 −27 1 1 Multiply the second equation by 2 to obtain Multiply the second row by 2 to obtain ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ y− 7 2 z = − 172 ⎣0 1 − 27 − 172 ⎦ 3y − 11z = −27 0 3 −11 −27 Add −3 times the second equation to the third Add −3 times the second row to the third to to obtain obtain ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ y − 27 z = − 172 ⎢0 1 − 27 − 172 ⎥ ⎣ ⎦ − 21 z = − 23 0 0 − 21 − 23 Multiply the third equation by −2 to obtain Multiply the third row by −2 to obtain ⎡ ⎤ x + y + 2z = 9 1 1 2 9 ⎢ ⎥ y − 27 z = − 172 ⎣0 1 − 27 − 172 ⎦ z= 3 0 0 1 3 Add −1 times the second equation to the ﬁrst Add −1 times the second row to the ﬁrst to to obtain obtain ⎡ ⎤ x + 11 z = 35 1 0 11 35 2 2 ⎢ 2 2 ⎥ y− 7 z = −2 17 ⎢0 1 − 27 17 ⎥ −2⎦ 2 ⎣ z= 3 0 0 1 3 The solution in this example Add −11 2 times the third equation to the ﬁrst Add − 112 times the third row to the ﬁrst and 7 2 can also be expressed as the or- and 27 times the third equation to the second to times the third row to the second to obtain dered triple (1, 2, 3) with the obtain ⎡ ⎤ x =1 1 0 0 1 understanding that the num- ⎢ ⎥ bers in the triple are in the y =2 ⎣0 1 0 2⎦ same order as the variables in z=3 0 0 1 3 the system, namely, x, y, z. The solution x = 1, y = 2, z = 3 is now evident. Exercise Set 1.1 1. In each part, determine whether the equation is linear in x1 , 2. In each part, determine whether the equation is linear in x x2 , and x3. and y. √ √ √ (a) x1 + 5x2 − 2 x3 = 1 (b) x1 + 3x2 + x1 x3 = 2 (a) 21/3 x + 3y = 1 (b) 2x 1/3 + 3 y = 1 π (c) x1 = −7x2 + 3x3 (d) x1−2 + x2 + 8x3 = 5 (c) cos x − 4y = log 3 (d) π cos x − 4y = 0 7 7 √ (e) xy = 1 (f ) y + 7 = x 3/5 (e) x1 − 2 x2 + x 3 = 4 (f ) πx1 − 2 x2 = 71/3 1.1 Introduction to Systems of Linear Equations 9 5 5 3. Using the notation of Formula (7), write down a general linear (a) , 87 , 1 (b) , 87 , 0 (c) (5, 8, 1) 7 7 system of 5 5 (a) two equations in two unknowns. (d) 7 , 107 , 2 7 (e) 7 , 227 , 2 (b) three equations in three unknowns. 11. In each part, solve the linear system, if possible, and use the (c) two equations in four unknowns. result to determine whether the lines represented by the equa- tions in the system have zero, one, or inﬁnitely many points of 4. Write down the augmented matrix for each of the linear sys- intersection. If there is a single point of intersection, give its tems in Exercise 3. coordinates, and if there are inﬁnitely many, ﬁnd parametric In each part of Exercises 5–6, ﬁnd a linear system in the un- equations for them. knowns x1 , x2 , x3 ,... , that corresponds to the given augmented (a) 3x − 2y = 4 (b) 2x − 4y = 1 (c) x − 2y = 0 matrix. 6x − 4 y = 9 4 x − 8y = 2 x − 4y = 8 ⎡ ⎤ ⎡ ⎤ 2 0 0 3 0 −2 5 ⎢ ⎥ ⎢ ⎥ 12. Under what conditions on a and b will the following linear 5. (a) ⎣3 −4 0⎦ (b) ⎣7 1 4 −3⎦ system have no solutions, one solution, inﬁnitely many solu- 0 1 1 0 −2 1 7 tions? 2x − 3y = a 0 3 −1 −1 −1 4x − 6y = b 6. (a) 5 2 0 −3 −6 In each part of Exercises 13–14, use parametric equations to ⎡ ⎤ describe the solution set of the linear equation. 3 0 1 −4 3 ⎢−4 −3 ⎥ 13. (a) 7x − 5y = 3 ⎢ 0 4 1 ⎥ (b) ⎢ ⎥ ⎣−1 3 0 −2 −9 ⎦ (b) 3x1 − 5x2 + 4x3 = 7 0 0 0 −1 −2 (c) −8x1 + 2x2 − 5x3 + 6x4 = 1 In each part of Exercises 7–8, ﬁnd the augmented matrix for (d) 3v − 8w + 2x − y + 4z = 0 the linear system. 14. (a) x + 10y = 2 7. (a) −2x1 = 6 (b) 6x1 − x2 + 3x3 = 4 3x1 = 8 5x2 − x3 = 1 (b) x1 + 3x2 − 12x3 = 3 9x1 = −3 (c) 4x1 + 2x2 + 3x3 + x4 = 20 (c) 2x2 − 3x4 + x5 = 0 (d) v + w + x − 5y + 7z = 0 −3x1 − x2 + x3 = −1 6x1 + 2x2 − x3 + 2x4 − 3x5 = 6 In Exercises 15–16, each linear system has inﬁnitely many so- lutions. Use parametric equations to describe its solution set. 8. (a) 3x1 − 2x2 = −1 (b) 2x1 + 2x3 = 1 15. (a) 2x − 3y = 1 4x1 + 5x2 = 3 3x1 − x2 + 4x3 = 7 6 x − 9y = 3 7x1 + 3x2 = 2 6x1 + x2 − x3 = 0 (b) x1 + 3x2 − x3 = −4 (c) x1 =1 3x1 + 9x2 − 3x3 = −12 x2 =2 −x1 − 3x2 + x3 = 4 x3 =3 9. In each part, determine whether the given 3-tuple is a solution 16. (a) 6x1 + 2x2 = −8 (b) 2x − y + 2z = −4 of the linear system 3x1 + x2 = −4 6x − 3y + 6z = −12 2x1 − 4x2 − x3 = 1 −4 x + 2 y − 4 z = 8 x1 − 3x2 + x3 = 1 In Exercises 17–18, ﬁnd a single elementary row operation that 3x1 − 5x2 − 3x3 = 1 will create a 1 in the upper left corner of the given augmented ma- trix and will not create any fractions in its ﬁrst row. (a) (3, 1, 1) (b) (3, −1, 1) (c) (13, 5, 2) 13 ⎡ ⎤ ⎡ ⎤ (d) 2 , 5 2 ,2 (e) (17, 7, 5) −3 −1 2 4 0 −1 −5 0 17. (a) ⎣ 2 −3 3 2⎦ (b) ⎣2 −9 3 2⎦ 10. In each part, determine whether the given 3-tuple is a solution 0 2 −3 1 1 4 −3 3 of the linear system ⎡ ⎤ ⎡ ⎤ x + 2y − 2z = 3 2 4 −6 8 7 −4 −2 2 3x − y + z = 1 18. (a) ⎣ 7 1 4 3⎦ (b) ⎣ 3 −1 8 1⎦ −x + 5y − 5z = 5 −5 4 2 7 −6 3 −1 4 10 Chapter 1 Systems of Linear Equations and Matrices In Exercises 19–20, ﬁnd all values of k for which the given Let x, y, and z denote the number of ounces of the ﬁrst, sec- augmented matrix corresponds to a consistent linear system. ond, and third foods that the dieter will consume at the main meal. Find (but do not solve) a linear system in x, y, and z 1 k −4 1 k −1 whose solution tells how many ounces of each food must be 19. (a) (b) 4 8 2 4 8 −4 consumed to meet the diet requirements. 3 −4 k k 1 −2 26. Suppose that you want to ﬁnd values for a, b, and c such that 20. (a) (b) the parabola y = ax 2 + bx + c passes through the points −6 8 5 4 −1 2 (1, 1), (2, 4), and (−1, 1). Find (but do not solve) a system 21. The curve y = ax 2 + bx + c shown in the accompanying ﬁg- of linear equations whose solutions provide values for a, b, ure passes through the points (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ). and c. How many solutions would you expect this system of Show that the coefﬁcients a , b, and c form a solution of the equations to have, and why? system of linear equations whose augmented matrix is ⎡ ⎤ 27. Suppose you are asked to ﬁnd three real numbers such that the x12 x1 1 y1 sum of the numbers is 12, the sum of two times the ﬁrst plus ⎢ 2 ⎥ ⎣x2 x2 1 y2 ⎦ the second plus two times the third is 5, and the third number x32 x3 1 y3 is one more than the ﬁrst. Find (but do not solve) a linear system whose equations describe the three conditions. y y = ax2 + bx + c True-False Exercises (x3, y3) TF. In parts (a)–(h) determine whether the statement is true or (x1, y1) false, and justify your answer. (a) A linear system whose equations are all homogeneous must (x2, y2) be consistent. x Figure Ex-21 (b) Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation. 22. Explain why each of the three elementary row operations does (c) The linear system not affect the solution set of a linear system. x− y =3 2x − 2y = k 23. Show that if the linear equations cannot have a unique solution, regardless of the value of k. x1 + kx2 = c and x1 + lx2 = d (d) A single linear equation with two or more unknowns must have the same solution set, then the two equations are identical have inﬁnitely many solutions. (i.e., k = l and c = d ). (e) If the number of equations in a linear system exceeds the num- 24. Consider the system of equations ber of unknowns, then the system must be inconsistent. ax + by = k cx + dy = l (f ) If each equation in a consistent linear system is multiplied through by a constant c, then all solutions to the new system ex + fy = m can be obtained by multiplying solutions from the original Discuss the relative positions of the lines ax + by = k , system by c. cx + dy = l , and ex + fy = m when (g) Elementary row operations permit one row of an augmented (a) the system has no solutions. matrix to be subtracted from another. (b) the system has exactly one solution. (h) The linear system with corresponding augmented matrix (c) the system has inﬁnitely many solutions. 2 −1 4 25. Suppose that a certain diet calls for 7 units of fat, 9 units of 0 0 −1 protein, and 16 units of carbohydrates for the main meal, and suppose that an individual has three possible foods to choose is consistent. from to meet these requirements: Working withTechnology Food 1: Each ounce contains 2 units of fat, 2 units of protein, and 4 units of carbohydrates. T1. Solve the linear systems in Examples 2, 3, and 4 to see how your technology utility handles the three types of systems. Food 2: Each ounce contains 3 units of fat, 1 unit of protein, and 2 units of carbohydrates. T2. Use the result in Exercise 21 to ﬁnd values of a , b, and c Food 3: Each ounce contains 1 unit of fat, 3 units of for which the curve y = ax 2 + bx + c passes through the points protein, and 5 units of carbohydrates. (−1, 1, 4), (0, 0, 8), and (1, 1, 7).

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