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8.1 MATRICES AND SYSTEMS OF EQUATIONS Copyright © Cengage Learning. All rights reserved. What You Should Learn • Write matrices and identify their orders. • Perform elementary row operations on matrices. • Use matrices and Gaussian elimination to solve systems of linear equations. • Use matrices and Gauss-Jordan elimination to solve systems of linear equations. 2 Matrices 3 Matrices A matrix having m rows and n columns is said to be of order m n. If m = n, the matrix is square of order m m (or n n). For a square matrix, the entries a11, a22, a33, . . . are the main diagonal entries. 4 Example 1 – Order of Matrices Determine the order of each matrix. a. b. Solution: a. This matrix has one row and one column. The order of the matrix is 1 1. b. This matrix has one row and four columns. The order of the matrix is 1 4. 5 Example 1 – Order of Matrices Determine the order of each matrix. c. d. Solution: c. This matrix has two rows and two columns. The order of the matrix is 2 2. d. This matrix has three rows and two columns. The order of the matrix is 3 2. 6 Matrices Row matrix: A matrix that has only one row Column matrix: Amatrix that has only one column Augmented matrix of the system: A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) Coefficient matrix of the system: the matrix derived from the coefficients of the system (but not including the constant terms) 7 Matrices System: x – 4y + 3z = 5 –x + 3y – z = –3 2x – 4z = 6 Augmented Matrix: Coefficient Matrix: 8 Elementary Row Operations 9 Elementary Row Operations 1. Interchange two equations. Interchange two rows 2. Multiply an equation by a nonzero constant. Multiply a row by a nonzero constant 3. Add a multiple of an equation to another equation. Add a multiple of a row to another row 10 Elementary Row Operations 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 11 Gaussian Elimination with Back-Substitution 12 Example 6 – Gaussian Elimination with Back-Substitution Solve the system y + z – 2w = –3 x + 2y – z = 2 . 2x + 4y + z – 3w = – 2 x – 4y – 7z – w = –19 Solution: Write augmented matrix. 13 Example 6 – Solution cont’d Interchange R1 and R2 so first column has leading 1 in upper left corner. Perform operations on R3 and R4 so first column has zeros below its leading 1. 14 Example 6 – Solution cont’d Perform operations on R4 so second column has zeros below its leading 1. Perform operations on R3 and R4 so third and fourth columns have leading 1’s. 15 Example 6 – Solution cont’d The matrix is now in row-echelon form, and the corresponding system is x + 2y – z = 2 y + z – 2w = –3 . z – w = –2 w= 3 Using back-substitution, you can determine that the solution is x = –1, y = 2, z = 1, and w = 3. 16 Gaussian Elimination with Back-Substitution 17 Gauss-Jordan Elimination 18 Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix. A second method of elimination, called Gauss-Jordan elimination, continues the reduction process until a reduced row-echelon form is obtained. 19 Example 8 – Gauss-Jordan Elimination Use Gauss-Jordan elimination to solve the system x – 2y + 3z = 9 –x + 3y = –4 . 2x – 5y + 5z = 17 Solution: The row-echelon form of the linear system can be obtained using the Gaussian elimination. 20 Example 8 – Solution cont’d Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows. Perform operations on R1 so second column has a zero above its leading 1. Perform operations on R1 and R2 so third column has zeros above its leading 1. 21 Example 8 – Solution cont’d The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, you have x=1 y = –1. z=2 Now you can simply read the solution, x = 1, y = –1, and z = 2 which can be written as the ordered triple(1, –1, 2). 22 Gaussian Elimination with Back substitution 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 Gaussian Jordon 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 23