Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form ax by h cx dy k where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero. Recall that the graph of each equation in the system is a straight line in the plane, so that geometrically, the solution to the system is the point(s) of intersection of the two straight lines L 1 and L 2, represented by the first and second equations of the system. Given the two straight lines L 1 and L 2, one and only one of the following may occur: 1. L 1 and L 2 intersect at exactly one point. Consistent Unique Solution ( 2. L 1 and L 2 are coincident. Dependent Infinitely many Solution 1 P a g e
3. L 1 and L 2 are parallel. Inconsistent No Solutions Solving Systems of Two Linear Equations in Two Variables To SOLVE a system of two linear equations in x and y means to find all of the ordered pairs whose coordinates make BOTH of the equations TRUE. You can solve a system by GRAPHING, SUBSTITUTION, or ELIMINATION (ADDITION). The SOLUTION SET may consist of a SINGLE POINT (CONSISTENT) or ALL THE POINTS ON A LINE (DEPENDENT), or there may be NO SOLUTION (INCONSISTENT). Example - A System of Equations With Exactly One Solution 1. 6x y 9 9x y 18 2. 2x y 1 3x 2y 12 3. 5x y 13 4x 2y 12 2 P a g e
Example - A System of Equations With Infinitely Many Solutions 1. 5x 5y 30 y x 6 2. 2x y 1 6x 3y 3 Example - A System of Equations With No Solutions 1. 2x y 1 6x 3y 12 2. 4x 2y 12 y 2x 10 3. 4x 14 2y 5y 10x 20 3 P a g e
Section 2.2 System of Linear Equations Unique Solutions The Gauss-Jordan Method The Gauss-Jordan elimination method is a technique for solving systems of linear equations of any size. The operations of the Gauss-Jordan method are 1. Interchange any two equations. 2. Replace an equation by a nonzero constant multiple of itself. 3. Replace an equation by the sum of that equation and a constant multiple of any other equation. Example Solve the following system of equations 2 x 4 y 6 z 22 3 x 8 y 5 z 27 x y 2 z 2 4 P a g e
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This rectangular array of 24 numbers, arranged in rows and columns and placed in red brackets, is an example of a matrix. The numbers inside the brackets are called elements of the matrix. Matrices are used to solve systems of linear equations. They give us a shortened way of writing a system of equations. The first step in solving a system of linear equations using matrices is to write the augmented matrix. An augmented matrix has a vertical bar separating the columns of the matrix into two groups. The coefficients of each variable are placed to the left of the vertical line and the constants are placed to the right. If any variable is missing, its coefficient is 0. Notice how the second matrix contains 1s down the diagonal from upper left to lower right, called the main diagonal, and 0s below the 1s. This arrangement makes it easy to find the solution of the system of equations with just a little back-substitution. 6 P a g e
Augmented Matrices Matrices are rectangular arrays of numbers that can aid us by eliminating the need to write the variables at each step of the reduction. For example, the system 2 x 4 y 6 z 22 3 x 8 y 5 z 27 x y 2 z 2 may be represented by the augmented matrix Every step in the Gauss-Jordan elimination method can be expressed with matrices, rather than systems of equations, thus simplifying the whole process: Write the augmented matrix of the given system of equations. 3x+4y=7 4x-2y=5 7 P a g e
Row-Reduced Form of a Matrix Each row consisting entirely of zeros lies below all rows having nonzero entries. The first nonzero entry in each nonzero row is 1 (called a leading 1). In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row. If a column contains a leading 1, then the other entries in that column are zeros. Row Operations 1. Interchange any two rows. 2. Replace any row by a nonzero constant multiple of itself. 3. Replace any row by the sum of that row and a constant multiple of any other row. Terminology for the Gauss-Jordan Elimination Method Unit Column A column in a coefficient matrix is in unit form if one of the entries in the column is a 1 and the other entries are zeros. Pivoting The sequence of row operations that transforms a given column in an augmented matrix into a unit column. Example - Pivot the matrix about the circled element 3 5 9 2 3 5 8 P a g e
Notation for Row Operations Letting R i denote the ith row of a matrix, we write Operation 1: R i R j to mean: Interchange row i with row j. Operation 2: cr i to mean: replace row i with c times row i. Operation 3: R i + ar j to mean: Replace row i with the sum of row i and a times row j. Write the system of equations corresponding to this augmented matrix. Then perform the row operation on the given augmented matrix. R 4r r 1 3 3 5 4 5 3 5 3 2 4 6 2 1 2 9 P a g e
Write the system of equations as an augmented matrix. Then perform this row operation on the given augmented matrix. R 4r r 3 1 3 x 3y z 5 4x 5y 3z 5 3x 2y 4z 6 The Gauss-Jordan Elimination Method 1. Write the augmented matrix corresponding to the linear system. 2. Interchange rows, if necessary, to obtain an augmented matrix in which the first entry in the first row is nonzero. Then pivot the matrix about this entry. 3. Interchange the second row with any row below it, if necessary, to obtain an augmented matrix in which the second entry in the second row is nonzero. Pivot the matrix about this entry. Continue until the final matrix is in row-reduced form. Gauss-Jordan Elimination Using Gaussian elimination we obtain a matrix in row-echelon form, with 1s down the main diagonal and 0s below the 1s. When you have accomplished this you must back-substitute to get your answers.a second method called Gauss-Jordan elimination continues the process until a matrix with 1s down the main diagonal and 0s in every position above and below each 1 is found. Such a matrix is said to be in reduced row-echelon form. For a system in three variables, x,y, and z, we must get the augmented matrix into the form seen below. 10 P a g e
Sometimes it is advantageous to write a matrix in reduced row echelon form. In this form, row operations are used to obtain entries that are 0 above as well as below the leading 1 in a row. The advantage is that the solution is readily found without needing to back-substitute. There will be a second advantage in the future when we discuss the inverse of a matrix. Graphing Calculator-Matrices, Reduced Row Echelon Form 1 To work with matrices we need to press 2nd x keys to get the Matrix Menu. To type in a matrix, cursor to the right twice to getto EDIT, press ENTER and type in the dimensions of matrix A. Press ENTER after each number, then type in the numbers that comprise the matrix, again pressing ENTER after each number. To get out of the matrix menu press QUIT (2nd MODE). 11 P a g e
To get the reduced row-echelon form bring up 1 the Matrix Menu (2nd x ) and cursor to MATH. Cursor down to B, rref (reduced row echelon form). Press ENTER. (Just row echelon form is "A:ref.") To type the name of the matrix again bring up the Matrix Menu and under NAMES, choose A #1, press ENTER, press ENTER again any you will have your new matrix. Example - Use the Gauss-Jordan elimination method to solve the system of equations 3 x 2 y 8 z 9 2 x 2 y z 3 x 2 y 3 z 8 12 P a g e
Solve this system of equations using matrices (row operations). 3x5y3 15x5y 21 Solve this system of equations using matrices (row operations). 2x y 4 2y 4z 0 3x 2z 11 Solve this system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. x y z 1 4x 3y 2z 16 2x 2y 3z 5 13 P a g e
Section 2.3 Systems of Linear Equations: Underdetermined A System of Equations with an Infinite Number of Solutions Example - Solve the system of equations given by x 2 y 3 z 2 3 x y 2 z 1 2 x 3 y 5 z 3 A System of Equations That Has No Solution Example - Solve the system of equations given by x y z 1 3 x y z 4 x 5 y 5 z 1 14 P a g e
Systems with no Solution If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line, then the system of equations has no solution. Theorem 1 a. If the number of equations is greater than or equal to the number of variables in a linear system, then one of the following is true: i. The system has no solution. ii. The system has exactly one solution. iii. The system has infinitely many solutions. b. If there are fewer equations than variables in a linear system, then the system either has no solution or it has infinitely many solutions. 15 P a g e
Section 2.4 Matrices A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has size m n. The entry in the ith row and jth column is denoted by a ij Notations for Matrices We can represent a matrix in two different ways. 1. A capital letter, such as A, B, or C, can denote a matrix. 2. A lowercase letter enclosed in brackets, such as that shown below can denote a matrix. A= a ij A general element in matrix A is denoted by a This 32 ij. th th refers to the element in the i row and j column. a is the element located in the 3rd row, 2nd column. See below. a11 a12 a13 a21 a22 a23 a31 a32 a 33 A matrix of order m n has m rows and n columns. If m=n, a matrix has the same number of rows as columns and is called a square matrix. 1 3 3 Let A= 4 5 3 3 2 4 a. What is the order of A? b. If A= a ij, identify a, and a 23 31 16 P a g e
Example - Applied Example: Organizing Production Data The Acrosonic Company manufactures four different loudspeaker systems at three separate locations. The company s May output is as follows: Model A Model B Model C Model D Location I 320 280 460 280 Location II 480 360 580 0 Location III 540 420 200 880 If we agree to preserve the relative location of each entry in the table, we can summarize the set of data as follows: 320 280 460 280 480 360 580 0 540 420 200 880 We have Acrosonic s May output expressed as a matrix: 320 280 460 280 P 480 360 580 0 540 420 200 880 What is the size (order) of the matrix P? Find a 24 (the entry in row 2 and column 4 of the matrix P) and give an interpretation of this number. Find the sum of the entries that make up row 1 of P and interpret the result. Find the sum of the entries that make up column 4 of P and interpret the result. 17 P a g e
Equality of Matrices Two matrices are equal if they have the same size and their corresponding entries are equal. Example - Solve the following matrix equation for x, y, and z [ ] [ ] Addition and Subtraction of Matrices If A and B are two matrices of the same size, then: 1. The sum A + B is the matrix obtained by adding the corresponding entries in the two matrices. 2. The difference A B is the matrix obtained by subtracting the corresponding entries in B from those in A. 18 P a g e
Example - Applied Example: Organizing Production Data The total output of Acrosonic for May is Model A Model B Model C Model D Location I 320 280 460 280 Location II 480 360 580 0 Location III 540 420 200 880 The total output of Acrosonic for June is Model A Model B Model C Model D Location I 210 180 330 180 Location II 400 300 450 40 Location III 420 280 180 740 Find the total output of the company for May and June Laws for Matrix Addition If A, B, and C are matrices of the same size, then A + B = B + A Commutative law (A + B) + C = A + (B + C) Associative law 19 P a g e
Subtract the following two matices. 3 1 2 1 0 4 4 3 Zero matrix = 0 0 0 0 3 2 3 2 Additive Inverses: The inverse of is 1 5 1 5 3 2 3 2 0 0 1 5 1 5 0 0 3 2 3 2 If A= then -A = 1 5 1 5 What is the zero matrix for all 23 matrices? What is the additive inverse for the matrix below? 3 1 0 4 20 P a g e
Transpose of a Matrix If A is an m x n matrix with elements a ij, then the transpose of A is the n xm matrix A T with elements a ji. 1 2 3 Example Transpose the following matrix A 4 5 6 7 8 9 Scalar Product If A is a matrix and c is a real number, then the scalar product ca is the matrix obtained by multiplying each entry of A by c. Example Given A 3 4 find the matrix X that satisfies 2X + B = 3A 3 2 1 2 and B 1 2 Example - Applied Example: Production Planning The management of Acrosonic has decided to increase its July production of loudspeaker systems by 10% (over June output). Find a matrix giving the targeted production for July. 21 P a g e
Section 2.5 - Multiplication of Matrices Multiplying a Row Matrix by a Column Matrix If we have a row matrix of size 1 n, A [ a 1 a 2 a 3 a n ] And a column matrix of size n 1, b 1 b 2 B b 3 b n Then we may define the matrix product of A and B, written AB, by b 1 b 2 AB [ a a a a ] b a b a b a b a b b n 1 2 3 n 3 1 1 2 2 3 3 n n Example Multiply the following two matrices 2 3 A [1 2 3 5] a n d B 0 1 22 P a g e
Dimensions Requirement for Matrices Being Multiplied Note from the last example that for the multiplication to be feasible, the number of columns of the row matrix A must be equal to the number of rows of the column matrix B. Dimensions of the Product Matrix From last example, note that the product matrix AB has size 1 1. This has to do with the fact that we are multiplying a row matrix with a column matrix. We can establish the dimensions of a product matrix schematically: More generally, if A is a matrix of size m n and B is a matrix of size n p, then the matrix product of A and B, AB, is defined and is a matrix of size m p. Schematically: The number of columns of A must be the same as the number of rows of B for the multiplication to be feasible Mechanics of Matrix Multiplication To see how to compute the product of a 2 3 matrix A and a 3 4 matrix B, suppose b b b b a a a 11 12 13 14 a11 a12 a13 A B b 21 b 22 b 23 b 24 a 21 a 22 a 23 b 31 b 32 b 33 b 34 we see that the matrix product C = AB is feasible (since the number of columns of A equals the number of rows of B) and has size 2 x 4. 23 P a g e
c 11 c 12 c 13 c 14 Thus C To see how to calculate the entries of C consider entry c 11 : c21 c22 c23 c 24 b 11 c 11 [ a11 a12 a13] b 21 a 11b 11 a 12b 21 a 13b 31 b 31 Matrix Multiplication Compute: 4 AB= 1 2 3 5 32 6 4 4 8 12 BA 5 1 2 3 5 10 15 6 6 12 18 24 P a g e
Example Compute AB 1 3 3 3 1 4 A B 4 1 2 1 2 3 2 4 1 Find the product 1 2 0 1 4 0 3 1 2 3 1 2 Find the product. 4 2 1 3 1 2 Find the product. 4 1 2 1 3 2 25 P a g e
Scalar Multiplication 2 3 6 3 If A= and B= 5 4 1 0 what is 2A+3B Find if possible 3A+B given that 1 2 1 0 A= and B= 3 1 2 3 26 P a g e
Laws for Matrix Multiplication If the products and sums are defined for the matrices A, B, and C, then (AB)C = A(BC) Associative law A(B + C) = AB + AC Distributive law Identity Matrix The identity matrix of size n is given by Properties of the Identity Matrix The identity matrix has the properties that I n A = A for any n r matrix A. BI n = B for any s n matrix B. In particular, if A is a square matrix of size n, then I A AI A n n Example Find and to see that they are the same value. 1 3 1 A 4 3 2 1 0 1 27 P a g e
Matrix Representation A system of linear equations can be expressed in the form of an equation of matrices. Consider the system 2 x 4 y z 6 3 x 6 y 5 z 1 x 3 y 7 z 0 The coefficients on the left-hand side of the equation can be expressed as matrix A below, the variables as matrix X, and the constants on right-hand side of the equation as matrix B: 2 4 1 x 6 A 3 6 5 X y B 1 1 3 7 z 0 The matrix representation of the system of linear equations is given by AX = B, or 2 4 1 x 6 3 6 5 y 1 1 3 7 z 0 To confirm this, we can multiply the two matrices on the left-hand side of the equation, obtaining 2 x 4 y z 6 3 x 6 y 5 z 1 x 3 y 7 z 0 which, by matrix equality, is easily seen to be equivalent to the given system of linear equations. 28 P a g e
Section 2.6 Inverse of a Square Matrix Let A be a square matrix of size n. A square matrix A 1 of size n such that 1 1 A A AA I n is called the inverse of A. Not every matrix has an inverse. A square matrix that has an inverse is said to be nonsingular. A square matrix that does not have an inverse is said to be singular. Example A Non-singular Matrix Show that the inverse of A 1 2 3 4 is 1 2 1 A 3 1 2 2 Example Show that the given matrix does not have an inverse B 0 1 0 0 Finding the Inverse of a Square Matrix Given the n x n matrix A: Adjoin the n x n identity matrix I to obtain the augmented matrix [A I ]. Use a sequence of row operations to reduce [A I ] to the form [I B] if possible. Then the matrix B is the inverse of A. 29 P a g e
Example Find the inverse of the given matrix 2 1 1 A 3 2 1 2 1 2 A Formula for the Inverse of a 2 x 2 Matrix a b Let A c d Suppose D = ad bc is not equal to zero. Then A 1 1 1 d b exists and is given by A D c a 30 P a g e
Example Find the inverse of the given A 1 2 3 4 Using Inverses to Solve Systems of Equations If AX = B is a linear system of n equations in n unknowns and if A 1 exists, then X = A 1 B is the unique solution of the system. Example Solve the system of Linear equations 2 x y z 1 3 x 2 y z 2 2 x y 2 z 1 31 P a g e