cKendra Kilmer January 11, 2008Section 2.1- Systems of Linear EquationsWhat is a linear equation?What does it mean to solve a system of linear equations?What are the possible cases when solving a system of linear equations?Example 1: Solve the following system of linear equations:4x − 5y = −302x + y = −8Example 2: Solve the following system of linear equations:2x − y = 16x − 3y = 12Example 3: Solve the following system of linear equations:3x − 7y = 46x − 14y = 81cKendra Kilmer January 11, 2008Note: We will first learn how to set-up the system in these word problems. We will then learn thebasics about matrices and then return to learn how to solve these problems.Example 4: A group of people are planning on taking a bus to an Aggie Football game in Austin.The bus holds 192 people. To be cost efficient, the leader plans on taking 192 people. Sincefreshmen are rowdy, there will be three times as many juniors on the trip as freshmen. If there arethe same number of sophmores on the trip as freshmen and juniors combined, find the number ofeach classification on the trip. (Note: The seniors won’t be going on the bus because they’re toocool.)Set-up the linear system:Example 5: An investment club has $200,000 earmarked for investment in stocks. To arrive atan acceptable overall level of risk, the stocks that management is considering have been classifiedinto three categories: high-risk, medium-risk, and low-risk. Management estimates that high-riskstocks will have a rate of return of 15%/year; medium-risk stocks, 10%/year; and low-risk stocks,6%/year. The members have decided that the investment in low-risk stocks should be equal to thesum of the investments in the stocks of the other two categories. Determine how much the clubshould invest in each type of stock if the investment goal is to have a return of $20,000/year on thetotal investment. (Assume that all of the money available for investment is invested.)Set-up the linear system:2cKendra Kilmer January 11, 2008Section 2.4 - MatricesDefinitions:1. A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has sizem × n. The entry in the ith row and jth column is denoted by ai j.2. A row matrix is a matrix of size 1 × n.3. A column matrix is a matrix of size m × 1.4. Two matrices are equal if they have the same size and equal corresponding entries.Example 1: Given the following matrix eqation, what are the values of a, b, and c?a 5−6 3b + 4=8 2c − 15−6 19Matrix Operations:1. Addition (You can only add matrices that have the same size)Example 2: Perform the following matrix operation:1 23 4+4 56 72. Subtraction (You can only subtract matrices that have the same size)Example 3: Perform the following matrix operation:1 23 4−4 56 73cKendra Kilmer January 11, 20083. Transpose: The transpose of an m × n matrix A with entries ai jis the n × m matrix ATwith entries aji.Example 4: Find ATifA =1 23 45 64. Scalar Product: For a matrix A and a real number c, the scalar product cA is found by multiplying eachentry in A by the real number c.Example 5: Find 3A ifA =1 23 45 6We can allow the calculator to do these basic calculations for us.(If you have a plain TI-83 (without the Plus), when you see directions to hit 2nd x−1, you need to hit theMATRIX button.)Enter the matrix into the calculator:• Hit 2nd x−1.• Cursor right two places to EDIT and hit ENTER on the matrix you wish to edit.• Enter the size of the matrix.• Enter the matrix elements.Call a matrix for a computation:• Make sure you are on the home screen before you begin any calculations. To do this, hit 2nd MODE toquit.• Press 2nd, x−1and cursor down under the NAMES list until you get to the matrix you want and hitENTER. The name of the matrix you need to do computations with will now be on your homescreen.Take the Transpose of a Matrix• Call the matrix you would like to transpose from the home screen.• Press 2nd, x−1, cursor right to MATH and select option 2:T.• You should now see the symbolic representation of transposing your matrix. To actually see the trans-pose, hit ENTER.4cKendra Kilmer January 11, 2008Example 6: Given the following matrices, perform the following operations on your calculator.A =4 56 92 3B =2 4 61 3 50 1 2C =1 23 45 6a) Find A + Cb) Find A − Bc) Find CTAugmented Matrix: We can combine two matrices into one, visually separating them by a verticalline. This is useful when solving a system of equations.Example 7:5cKendra Kilmer January 11, 2008Section 2.2 - Systems of Linear Equations (The Gauss-Jordan Method)This method allows us to strategically solve systems of linear equations. We perform operations onan augmented matrix that is formed by combining the coefficient matrix and the constant matrix asshown in the next example.Example 1: Find the intial augmented matrix for the system of equations below:a) 2x − 4y = 10y = 1 − 3xb) x1− 2x2= 10x3+ 58x2= x1− 3x34x1− 3x3= x2The goal of the Gauss-Jordan Elimination Method is to get the augmented matrix into RowReduced Form. A matrix is in Row Reduced Form when:1. Each row of the coefficient matrix consisting entirely of zeros lies below any other row having nonzeroentries.2. The first nonzero entry in each row is 1 (called a leading 1)3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1 inthe upper row.4. If a column contains a leading 1, then the other entries in that column are zeros.Note: We only consider the coefficient side (left side) of the augmented matrix when determiningwhether the matrix is in row-reduced form.Example 2: Are the following in Row Reduced form?a)1 0 0 00 1 0 00 0 1 3b)1 2 0 00 0 1 00 0 0 1c)1 2 0 00 0 1 30 0 2 16cKendra Kilmer January 11, 2008To put a matrix in Row Reduced Form, there are three valid Row Operations:1. Interchange any two rows (Ri↔ Rj)2. Replace any row by a nonzero constant multiple of itself (cRi)3. Replace any row by the sum of that row and a constant multiple of any other row (Ri+ cRj).Steps for Gauss Jordan Elimination:1. Begin by transforming the top left corner element, a11, into 1. This is your first pivot element.2. Next, transform the other elements in its column into zeros using the 3 row operations.3. Choose the next pivot element (diagonal down from the first pivot