[ ] 1 [ B] 7.2 Matrix Algebra Pre Calculus. 7.2 MATRIX ALGEBRA (Day 1)

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1 7. Matrix Algebra Pre Calculus 7. MATRIX ALGEBRA (Day 1) Learning Targets: 1. Use matrices to represent data.. Add or Subtract matrices without a calculator. 3. Multiply a scalar by a matrix without a calculator. 4. Multiply matrices without a calculator. Matrix: a rectangular array of numbers used to record data. Each number in a matrix, called an, has a distinct position according to its row # and column #. The number of rows and columns determines the of the matrix. Exploration: After completing a gourmet foods class, Theresa, Marita and William have started a small business making and selling candy to specialty stores. After researching the market they decided to make three kinds of candy: toffee, fudge and caramels. They decided to use the kitchens in their homes and make the candy independently because of their conflicting schedules. To minimize their cost they purchased ingredients in bulk quantities. 1) In the first month of production, Theresa made lbs of toffee, lbs of fudge and lbs of caramels. Marita made lbs of toffee, 4 lbs of fudge and 3 lbs of caramels. William made 5 lbs of toffee, lbs of fudge and 3 lbs of caramels. Design an initial matrix for the number of lbs of each type of candy made by the three gourmet confectioners. Call this matrix[ M 1 ]. [ M ] 1 Theresa = Marita William toffee fudge caramels ) The next month, each of the three confectioners increased their production by 50%. Create a matrix to represent the production in the second month. Call this matrix[ M ]. [ M 1 ] multiplied by what scalar produces [ M ]? 3) In the third month, after refining their candy making production each of the three confectioners doubled their initial month s production. Create a matrix to represent the production in the third month. Call this matrix[ M ]. 3 M multiplied by what scalar produces[ M ]? [ ] 1 3 4) At the end of the first quarter, that is the first three months of production and sales, the confectioners analyzed the numbers. First they looked at total production. Create a matrix to represent the total lbs of candy by type that the three produced. Call this matrix[ A ] 5) How many lbs of toffee, fudge and caramels did the three produce? 6) Sugar, butter and chocolate are the ingredients required to make the candy. Toffee requires ¾ of a lb of sugar, ¼ of a lb of butter. Fudge requires ¾ of a lb of sugar, 3/0 of a lb of chocolate, and 1/10 of a lb of butter. Caramels require 4/5 of a lb of sugar, 3/0 of a lb of chocolate, and 1/0 of a lb of butter. Create a matrix indicating the quantity of ingredients necessary to make each type of candy. Call this matrix[ B ] [ B] = toffee fudge caramels sugar butter chocolate Unit 9-1

2 7. Matrix Algebra Pre Calculus 7) Theresa would like to know how many lbs of sugar, butter and chocolate she has used to make all of the candy she s made for the first three months. Write an expression using values from the matrices to calculate for Theresa how many a) lbs of sugar she used. b) lbs of butter she used. c) lbs of chocolate she used. 8) In order to purchase their next bulk order of ingredients, all three gourmet confectioners would like to determine how much of each of the three ingredients they each used in their first three months of production. Show what two matrices and what operation will be useful to create a matrix that represents the lbs of ingredients used by the three confectioners. Call the result matrix[ E ] 9) Ingredients cost $1.89 per lb for sugar, $.65 per lb for butter and $3.8 per lb for chocolate. Create a matrix to represent the cost of the ingredients per lb. Call this matrix [ C ] a) Find the cost of making one lb of toffee b) Find the cost of making one lb of fudge [ C] = sugar butter choclate Cost c) Find the cost of making one lb of caramels d) Show what two matrices and what operation will be useful in calculating this information? Call the result of this multiplication matrix[ D ] 10) If the three gourmet confectioners decide to contribute to the cost of purchasing the ingredients based upon how many lbs of each ingredient they used in production. a) How much does Theresa need to contribute for the first quarter cost of ingredients? b) Show what operation and what two matrices will be useful in calculating this information for all three of the gourmet confectioners? c) How much does Marita contribute for the first quarter cost of ingredients William? d) Show an alternative set of matrices that we could use to calculate the same result from part (b). Unit 9-

3 7. Matrix Algebra Pre Calculus The gourmet candy maker example illustrates the usefulness of matrices for organizing data and working with data. You will be responsible for working with matrices algebraically using paper and pencil and also using the matrix feature on your calculator more with the calculator next class. To review what we learned, use the matrices below in the following examples [ A] = [ B] = [ C] = 1 4 [ D] = Example 1: The order of a matrix is indicated by the and. R x C cola [ A] and [ B ] are x matrices. [ C ] is [ D ] is Example : A is a number that will scale all of the elements in a matrix by a factor. Find 3[ A ] = [ ] 1 D = Example 3: Adding or subtracting matrices can only be done on matrices with the. Perform the operations or explain why the operation cannot be completed. [ A] [ B] = [ A] 3[ C] + = Is matrix addition commutative? Example 4: Multiplying matrices requires that the number of columns in the first matrix be equal to the number of rows in the second matrix. The solution to matrix multiplication will have the outer dimensions of the matrices being multiplied. [ A][ B ] = [ B][ A ] = [ A][ D ] = [ C][ D ] = Is matrix multiplication commutative? Unit 9-3

4 7. Matrix Algebra Pre Calculus 7. MATRIX ALGEBRA (Day ) Warm Up: 1. Complete the following: a) 4 + = 4 b) 5 = 5 c) 3x + = 3x d) 8, = 8,. Describe to a classmate the two basic questions above how are they alike?...how are they different? 3. Explain how you would solve each of the following: a) 3x = 1 b) x = 5 c) x = d) x e = 7 e) 4 x = 0 4. Be ready to discuss as a class how the questions in #3 use the concepts in question #1. Learning Targets: 1. Find and use the identity matrix.. Calculate the determinant of a x matrix and use it to determine whether a x matrix has an inverse. 3. Find the inverse of a matrix using a calculator. 4. Use a calculator to perform all matrix operations: addition, subtraction, scalar & matrix multiplication, finding determinants & finding inverses. Before we proceed with our understanding and use of matrices, we need to review some mathematical terms from Algebra that you may have forgotten. These terms were illustrated in our warm-up. 1.. So are there matrices that do the same thing? Example 1: Try this a) = 0 0 b) = 0 1 Identity Matrix of order n x n: Matrix with on the main diagonal (upper left to lower right) and everywhere else. Inverse of a Square Matrix Let [ A] be an n x n square matrix. If there is a matrix [ B ] such that [ A][ B] = [ B][ A] = [ I], then matrix [ B ] is the inverse of [ A ]. We use inverse notation: If a matrix has no inverse, we call it. If a matrix HAS an inverse, we call it. Unit 9-4

5 7. Matrix Algebra Pre Calculus Example : Decide whether or not the given matrices are inverses. [ A] 6 = B = 4 1 [ ] Deciding whether two given matrices are inverses is simple when compared to the task of determining whether a matrix is singular or non-singular. Without two matrices to check, we need a better way than to try and think about undoing matrix multiplication a b Determinant: Given matrix [ A] = c d, the determinant is ad bc. The notation det [ A ] is used to indicate a determinant. If the determinant is, a matrix is singular. If the determinant is any other number, the matrix is. We will restrict our study of determinants to x ordered matrices. Interestingly the determinant also helps us FIND the inverse of a matrix Inverse of a x Matrix If ad bc 0, then 1 a b 1 d b = c d ad bc c a Example 3: Given the following matrices: 3 [ A] = [ B] = 8 [ C] 1 1 = a) Determine whether [ A ] and [ B] are singular or non-singular matrices. For any non-singular matrices, find the inverse. b) Use your calculator to find [ ] 1 C. Unit 9-5

6 7.3 Multivariate Linear Systems Pre Calculus 7.3 MULTIVARIATE LINEAR SYSTEMS Learning Targets: 1. Review solving systems of equations (algebraically and graphically).. Extend systems of linear equations to include more than two variables. 3. Solve matrix equations using inverses. 4. Apply process of solving matrices to application problems. System of Equations: a set of equations defined by the same variables. The Solution of a System is a value for each variable that satisfies equations (aka makes them true). o o No Solution means: Infinitely Many Solutions means: Linear Systems are usually solved by or. Non-Linear Systems are usually solved by or. Example 1: Identify a strategy to solve each system below. Then solve it. a) Use: b) Use: c) Use: x y = 10 3x+ y = 1 y = 6 x 3x y = y = ln x y = x 4x+ But what if we have more variables? Then we need more equations! At this point we will restrict our studies to linear systems, but we can extend what we know from Algebra to include linear systems with MORE than two variables. Example : Use algebra to solve the system below. x y+ z = 7 y z = 7 z = 3 Triangular Form: a form of a system that is equivalent to a given system, but where the solution is easy to read. Transforming a system to a triangular system is called, named for the famous German mathematician. The following operations produce an equivalent system of linear equations. 1. Interchanging any two equations in a system.. Multiplying (or dividing) one equation by a nonzero real number. 3. Adding a multiple of one equation to any other equation in the system. Unit 9-6

7 7.3 Multivariate Linear Systems Pre Calculus Example 3: Use Gaussian elimination to solve the system below. x y z = 3 3x 5y+ z = 14 x y+ z = 7 So what does all this have to do with Matrices? We can solve a system with matrices if the number of is equal to the number of AND the coefficient matrix is. Example 4: Set up a matrix equation for the system. Then, solve. a) 81.x+ 17.4y = x 1.8y = 1.8 b) a b+ c = 5 4a+ b+ c = 1 9a+ 3b+ c = 13 Notice in example 4 how easy a matrix equation is to solve even if it has more than two variables! We can use a system of linear equations and matrices to find a function rule for a polynomial when given ordered pairs of points that lie on that polynomial. Example 5: Find the equation of the quadratic function so that points ( 1, 5), (, 1) and (3, 13) are its graph. Unit 9-7