6.3 Notes O Brien F15

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1 CA th ed HL. Notes O Brien F. Solution of Linear Systems by ow Transformations I. Introduction II. In this section we will solve systems of first degree equations which have two or more variables. We will use matrices and three row operations to rewrite the system into reduced row-echelon form. Example: x y z = 8 8 x z = 9 9 x y z = Terminology A. Matrix A matrix is a rectangular array of numbers. The plural of matrix is matrices. Example: B. Entry 9 A horizontal line of numbers is called a row. A vertical line of numbers is called a column. Each number or element in a matrix is called an entry. Each entry in a matrix can be designated by a letter with two subscripts which indicate the row and column it is in. C. Order Example: a, is the entry in the first row, second column. In the matrix above, a, =. A matrix which has m rows and n columns is said to be of order m x n (read m by n ). When giving the order of a matrix, the number of rows always comes first, followed by the number of columns. Example: The matrix shown above has rows and columns, so it is a x matrix. D. Main Diagonal Entries a,, a,,..., a n, n form the main diagonal of the matrix. Example: In the matrix above, the main diagonal contains entries a, =, a, =, and a, = E. educed ow-echelon Form of a Matrix A matrix in reduced row-echelon form (EF) has the following properties:. All rows consisting entirely of zeros occur at the bottom of the matrix.. For each row that does not consist entirely of zeros, the first non-zero entry is (the leading ).. For two successive non-zero rows, the leading in the lower row is farther right than the leading in the higher row.. Each column that contains a leading has zeros above and below the leading.

2 CA th ed HL. Notes O Brien F F. Augmented Matrix An augmented matrix is written using the coefficients and constants of a system of linear equations. We use a vertical bar to separate the coefficients from the constants. If a term is missing from an equation, we insert zero as a placeholder in the matrix. x y z = 9 x z = x y z = 9 III. IV. ow Operations To rewrite a system of equations into reduced row-echelon form, we can use three basic row operations to produce equivalent systems (systems which have exactly the same solution as the original).. Interchange two rows [Swap rows] Example of notation: This operation is used to get a in a pivot position.. Multiply each entry in a row by a nonzero constant [Multiply] Example of notation: This operation is also used to get a in a pivot position, usually by multiplying the row by the reciprocal of the leading coefficient.. Add a multiple of one row to another [Pivot] Example of notation: This operation is used to get a above or below a one in a pivot position. Typically we take (the opposite of the leading coefficient x the pivot row) the row we are changing. Caution: When we add a multiple of a row to another row, only the row we add to changes. The row we multiply does not change. It is called the pivot row. The Sequence of Operations in Gauss-Jordan Elimination Our goal is to rewrite the given system of equations into reduced row-echelon form. To do this, we work from left to right, transforming one column at a time. First we get a one in the pivot position and then we get zeros above and below the one. Starting with the first column:. Get a in the pivot position using owop (Swap) or owop (Multiply). Be sure you record your row operation in proper form. (Example: ).. Get s above and below the pivot by using owop (Pivot). Be sure you record your row operations in proper form. (Example: ). epeat steps and for each column, working from left to right. Always get the in the pivot position and then s above and below it before you move to the next column. Once you have reduced row-echelon form (s along main diagonal and s above and below), write your solution as an ordered pair or ordered triple (i.e., as a point). You can check your solution by plugging it into each equation in the original system of equations or by using SOLVSYS or SOLVSYS. Hint: The first pivot is in row, column. The second pivot is in row, column. The third pivot is in row, column. The nth pivot is in row n, column n. Thus the pivot moves along the main diagonal.

3 CA th ed HL. Notes O Brien F V. OWOPS Program. Before you enter the program, you must enter your matrix as Matrix A. Go to the MATIX menu. ight arrow two times to EDIT. Select [A]. Enter # of rows followed by the # of columns. Enter matrix, row by row, from left to right. Hit enter after each entry. Before you leave this screen, check your matrix for errors by using the arrow keys to move from column to column, row to row.. Turn the program OWOPS on. Start on a blank line on the home screen. Hit the Program key. Select the number of the program called OWOPS. You should see prgmowops. Hit enter. You should see the matrix you just entered. ecord the initial matrix on your paper. Hit enter again to advance the program.. Performing ow Operations. Hints A. To get a one in a pivot position by swapping rows: Select TO SWAP OWS. Enter the number of the first row you want to swap. Hit enter. Enter the number of the other row you want to swap. Hit enter. ecord the row operation in proper form (Ex: ). ecord the resulting matrix. B. To get a one in a pivot position by multiplying a row: Select TO MULTIPLY. Enter the number of the row you want to multiply. Hit enter. Enter the number you want to multiply by (usually the reciprocal of the leading coefficient). Hit enter. ecord the row operation in proper form (Ex: ¼ ). ecord the resulting matrix. C. To get a zero above or below a one in a pivot position: Select TO PIVOT. Enter the row number where the pivot is located. Hit enter. Enter the column number where the pivot is located. Hit enter. ecord the row operation in proper form (Ex: ). ecord the resulting matrix. D. To exit the OWOPS program: Select TO STOP. Hit enter. Hit clear. You will not use owop (Swap) very often. In fact, you can do all Gauss-Jordan Elimination problems using just owop (Multiply) and owop (Pivot). When you use owop (Pivot), the program will get zeros in every position in the column except where the pivot is located. Be sure you write down the appropriate row operation to get each zero. You must record every matrix and every row operation. It is o.k. to record more than one pivoting operation on a single matrix, but do not record the operations to get zeros on the same matrix where you record the operation to get a one. Pivot C is not the proper form for recording an operation to get zeros. You must be specific about what you multiplied the pivot row by and what row you added that to.

4 CA th ed HL. Notes O Brien F VI. The Number of Solutions of a Linear System In a three dimensional coordinate system, the graph of a linear equation in three variables is a plane.. A consistent, independent system would mean the three planes were intersecting in one point, like the corner of a room. A solution to this type of system is an ordered triple such as (,, ).. A consistent, dependent system would mean that the three planes were intersecting in a line or that all three were the same plane. A solution to this type of system would look like (z, z, z) where two of the variables (usually x & y) are expressed in terms of the third (usually z). Note the all zero row.. An inconsistent system would mean that the three planes have no points of intersection common to all three. The solution to this type of system will be the empty set. Note the row with three zeros and a non-zero constant. VII. Sample Problems Example Solve the following system of equations using Gauss-Jordan Elimination with Matrices. x y = x y = (like on p. ) Solution: (, ) Example Solve the following system of equations using Gauss-Jordan Elimination with Matrices. x y z = x y z = x y z = (like 8 on p. ) 9 Solution: (,, )

5 CA th ed HL. Notes O Brien F Example Solve the following system of equations using Gauss-Jordan Elimination with Matrices. x y = 8x y = (like 8 on p. ) No Solution Example Solve the following system of equations using Gauss-Jordan Elimination with Matrices. x y z = x y 8z = x y 8z = (like 9 on p. ) x z y = z = 8 z = z x = z y = z 8 z = z Solution: ( z, z 8, z) Example Solve the following application problem by using Gauss-Jordan Elimination with Matrices. Pat Summers wins $, in the Louisiana state lottery. He invests part of the money in real estate with an annual return of % and another part in a money market account at.% interest. He invests the rest, which amounts to $8, less than the sum of the other two parts, in certificates of deposit that pay.%. If the total annual interest on the money is $9, how much was invested at each rate? x = money invested in real estate at % y = money invested in money market account at.% z = money invested in CDs at.% (like 9 on p. ) total invested was $, x y z =, total interest income was $,9.x.y.z =,9 $8, less than the sum of the other two parts, in certificates of deposit z = x y 8, x y z =,.x.y.z =,9 x y z = 8, x y z = 8,

6 CA th ed HL. Notes O Brien F x y z =,.x.y.z =,9 x y z = 8, Solution: $, is invested in real estate at % $, is invested in money market account at.% $, is invested in CDs at.%