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Slide 9.1- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Matrices and Systems of Equations Learn the definition of a matrix. Learn to use matrices to solve a system of linear equations. Learn to use the Gaussian elimination procedure. Learn to use the Gauss-Jordan elimination procedure. SECTION 9.1 1 2 3 4

Slide 9.1- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A MATRIX A matrix is a rectangular array of numbers denoted by Row 1 Row 2 Row m Column 1Column 2Column n

Slide 9.1- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A MATRIX If a matrix A has m rows and n columns, then A is said to be of order m by n (written m 5 n). The entry or element in the ith row and jth column is a real number and is denoted by the double-subscript notation a ij. The entry a ij is sometimes referred as the (i, j)th entry or the entry in the (i, j) position, and we often write

Slide 9.1- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A MATRIX We also write A mn to indicate that the matrix A has m rows and n columns. If m = n then A is called a square matrix of order n and is denoted by A n. The entries a 11, a 22, …, a nn form the main diagonal of A n. A 1 5 n matrix is called a row matrix, and an n 5 1 matrix is called a column matrix.

Slide 9.1- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MATRIX AND LINEAR SYSTEMS We can display all the numerical information contained in a linear system in an augmented matrix of the system. Coefficients of z Coefficients of y Coefficients of x Constants

Slide 9.1- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ELEMENTARY ROW OPERATIONS Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Row OperationIn SymbolsDescription Interchange two rows R i R j Interchange the ith and jth rows Multiply a row by a nonzero constant cR j Multiply the jth row by c. Add a multiple of one row to another row cR i + R j  R j Replace the jth row by adding c times jth row to it.

Slide 9.1- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ROW-ECHELON FORM AND REDUCED ROW-ECHELON FORM An m 5 n matrix is in row-echelon form if it has the following three properties: 1.All nonzero rows are above the rows consisting entirely of zeros. 2.The leading entry of each nonzero row is 1. 3.For two successive rows, the leading 1 in the higher row is farther to the left of the leading 1 in the lower row.

Slide 9.1- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ROW-ECHELON FORM AND REDUCED ROW-ECHELON FORM If a matrix in row-echelon form has the following additional property, then it is in reduced row-echelon form: 4.Each leading 1 is the only nonzero entry in its column.

Slide 9.1- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR SOLVING LINEAR SYSTEMS BY USING GAUSSIAN ELIMINATION Step 1.Write the augmented matrix. Step 2.Use elementary row operations to transform the augmented matrix into row-echelon form. Step 3.Write the system of linear equations that correspond to the matrix in row-echelon form that was obtained in Step 2. Step 4.Use the system of equations obtained in Step 3, together with back-substitution, to find the solution set of the system.

Slide 9.1- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solve the system of equations by using Gaussian elimination. Solution Step 1 The augmented matrix of the system is.

Slide 9.1- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solution continued Step 2

Slide 9.1- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solution continued Step 2

Slide 9.1- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solution continued Step 2 This is in row- echelon form. Step 3The system corresponding to the last matrix in Step 2 is

Slide 9.1- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solution continued Back-substitute z = 3 and y = –1 in Equation 1. Step 4Equation (3) in Step 3 gives the value z = 3. Back-substitute z = 3 in Equation (2).

Slide 9.1- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a System by Using Gaussian Elimination Solution continued The solution set for the system is {(2, –1, 3)}. You should check the solution by substituting these values for x, y, and z into the original system of equations.

Slide 9.1- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a System of Equations by Gauss- Jordan Elimination Solve the system given in Example 4 by Gauss- Jordan elimination. Recall that the given system is Solution The augmented matrix of the system is.

Slide 9.1- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a System of Equations by Gauss- Jordan Elimination Solution continued

Slide 9.1- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a System of Equations by Gauss- Jordan Elimination Solution continued

Slide 9.1- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a System of Equations by Gauss- Jordan Elimination Solution continued

Slide 9.1- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a System of Equations by Gauss- Jordan Elimination Solution continued We now have an equivalent matrix in reduced row-echelon form. The corresponding system of equations for the last augmented matrix is: Hence, the solution set is {(5, 1, 3)}, as in Example 4.

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