Summary 309 Solving Systems of Linear Equations by Using Matrices Key Concepts A matrix is a rectangular array of numbers displayed in rows and columns. Every number or entry within a matrix is called an element of the matrix. The order of a matrix is determined by the number of rows and number of columns.A matrix with m rows and n columns is an m n matrix. A system of equations written in standard form can be represented by an augmented matrix consisting of the coefficients of the terms of each equation in the system. The Gauss-Jordan method can be used to solve a system of equations by using the following elementary row operations on an augmented matrix. 1. Interchange two rows. 2. Multiply every element in a row by a nonzero real number. 3. Add a multiple of one row to another row. These operations are used to write the matrix in reduced row echelon form. c1 0 0 1 ` a b d This represents the solution, x a and y b. Examples Example 1 1 2 5 is a 1 3 matrix (a row matrix). c 2 2 1 is a matrix (a square matrix). 1 8 5 d c 2 1 4 1 is a matrix (a column matrix). d Example 2 The augmented matrix for is 4x y 12 x 2y 6 Example 3 Solve the system from Example 2 by using the Gauss- Jordan method. c 1 2 4 1 R13R2 d c 1 2 0 9 ` 6 12 4R1 R2 1 R2 d c 1 2 0 1 ` 6 36 d 19 c1 0 0 1 ` 6 4 ` 2 4 d R2 1 R2 2R2 R1 1 R1 From the reduced row echelon form of the matrix we have x 2 and y 4. The solution set is 512, 426. Section 3.7 c4 1 1 2 ` 12 6 d
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