Copyright © 2011 Pearson Education, Inc. Solving Linear Systems Using Matrices Section 6.1 Matrices and Determinants

6.1 Copyright © 2011 Pearson Education, Inc. Slide 6-3 A matrix is a rectangular array of real numbers. The rows of a matrix run horizontally, and the columns run vertically. A matrix with only one row is a row matrix or row vector. A matrix with only one column is a column matrix or column vector. A matrix with m rows and n columns has size m  n (read “m by n”). The number of rows is always given first. A square matrix has an equal number of rows and columns. Each number in a matrix is called an entry or an element. The matrix of coefficients for a system of equations in standard form, respectively, is the coefficient matrix for the system. The constants from the right-hand side of the system are attached to the matrix of coefficients, to form the augmented matrix of the system. Matrices

6.1 Copyright © 2011 Pearson Education, Inc. Slide 6-4 Any of the following row operations on an augmented matrix gives an equivalent augmented matrix: 1. Interchanging two rows of the matrix Abbreviated R i  R j (interchange rows i and j) 2. Multiplying every entry in a row by the same nonzero real number Abbreviated aR i  R i (a times row i replaces row i) 3. Adding to a row a nonzero multiple of another row Abbreviated aR i + R j  R j (aR i + R j replaces R j ) Summary: Row Operations

6.1 Copyright © 2011 Pearson Education, Inc. Slide 6-5 The goal of the Gaussian elimination method is to convert the coefficient matrix (in the augmented matrix) into an identity matrix using row operations. The diagonal of a matrix consists of the entries in the first row first column, second row second column, third row third column, and so on. A square matrix with ones on the diagonal and zeros elsewhere is an identity matrix. If the system has a unique solution, then it will appear in the rightmost column of the final augmented matrix. Definitions

6.1 Copyright © 2011 Pearson Education, Inc. Slide 6-6 To solve a system of two linear equations in two variables using Gaussian elimination, perform the following row operations on the augmented matrix. 1. If necessary, interchange R 1 and R 2 so that R 1 begins with a nonzero entry. 2. Get a 1 in the first position on the diagonal by multiplying R 1 by the reciprocal of the first entry in R Add an appropriate multiple of R 1 to R 2 to get 0 below the first Get a 1 in the second position on the diagonal by multiplying R 2 by the reciprocal of the second entry in R Add an appropriate multiple of R 2 to R 1 to get 0 above the second Read the unique solution from the last column of the final augmented matrix. Procedure: The Gaussian Elimination Method for an Independent System of Two Equations

6.1 Copyright © 2011 Pearson Education, Inc. Slide 6-7 A system is independent if it has a single solution. The coefficient matrix of an independent system is equivalent to an identity matrix. A system is inconsistent if it has no solution and is dependent if it has infinitely many solutions. Applying Gaussian elimination to an inconsistent system causes a row to appear with 0 as the entry for each coefficient but a nonzero entry for the constant. For a dependent system of two equations in two variables, a 0 will appear in every entry for some row. Likewise, for a system of three equations in three variables we have the same results for inconsistent and dependent systems. Inconsistent and Dependent Equations