892 CHAPTER 9 Systems and Matrices 9.2 Matrix Solution of Linear Systems ■ The Gauss-Jordan Method ■ Special Systems ■ The Gaussian Elimination Method Systems of linear equations occur in many practical situations, and as a result, computer methods have been developed for efficiently solving linear systems. Computer solutions of linear systems depend on the idea of a matrix (plural matrices), a rectangular array of numbers enclosed in brackets. Each number is an element of the matrix. Linear system of equations x + 3y + 2z = 1 2x + y - z = 2 can be written as x + y + z = 2 Augmented matrix £1 2 1 3 1 1 2 -1 1 3 1 2 2§ Rows Columns The vertical line, which is optional, separates the coefficients from the constants. Because this matrix has 3 rows (horizontal) and 4 columns (vertical), we say its dimension* is 3 * 4 (read “three by four”). The number of rows is always given first. To refer to a number in the matrix, use its row and column numbers. For example, the number 3 is in the first row, second column. We can treat the rows of this matrix just like the equations of the corresponding system of linear equations. Because an augmented matrix is nothing more than a shorthand form of a system, any transformation of the matrix that results in an equivalent system of equations can be performed. Matrix Row Transformations For any augmented matrix of a system of linear equations, the following row transformations will result in the matrix of an equivalent system. 1. Interchange any two rows. 2. Multiply or divide the elements of any row by a nonzero real number. 3. Replace any row of the matrix by the sum of the elements of that row and a multiple of the elements of another row. These transformations are restatements in matrix form of the transformations of systems discussed in the previous section. From now on, when referring to the third transformation, we will abbreviate “a multiple of the elements of a row” as “a multiple of a row.” Before matrices can be used to solve a linear system, the system must be arranged in the proper form, with variable terms on the left side of the equation and constant terms on the right. The variable terms must be in the same order in each of the equations. * Other terms used to describe the dimension of a matrix are order and size. J2 3 7 5 -1 10R Matrix The Gauss-Jordan Method In this section, we develop a method for solving linear systems using matrices. We start with a system and write the coefficients of the variables and the constants as an augmented matrix of the system.
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