SECTION 11.2 Systems of Linear Equations: Matrices 773 4 Solve a System of Linear Equations Using Matrices To solve a system of linear equations using matrices, use row operations on the augmented matrix of the system to obtain a matrix that is in row echelon form . DEFINITION Row Echelon Form A matrix is in row echelon form when the following conditions are met: • The entry in row 1, column 1 is a 1, and only 0’s appear below it. • The first nonzero entry in each row after the first row is a 1, only 0’s appear below it, and the 1 appears to the right of the first nonzero entry in any row above. • Any rows that contain all 0’s to the left of the vertical bar appear at the bottom. For example, for a system of three equations containing three variables, x y , , and z, with a unique solution, the augmented matrix is in row echelon form if it is of the form ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ a b c d e f 1 0 0 1 0 1 where a b c d e ,,, ,,and f are real numbers. The last row of this augmented matrix states that = z f. We then determine the value of y using back-substitution with = z f, since row 2 represents the equation + = y cz e. Finally, x is determined using back-substitution again. Two advantages of solving a system of equations by writing the augmented matrix in row echelon form are the following: • The process is algorithmic; that is, it consists of repetitive steps that can be programmed on a computer. • The process works on any system of linear equations, no matter how many equations or variables are present. The next example shows how to solve a system of linear equations by writing its augmented matrix in row echelon form. How to Solve a System of Linear Equations Using Matrices (Row Echelon Form) Solve: + = + + = + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) x y x y z x y z 2 2 6 1 3 4 13 EXAMPLE 5 Step-by-Step Solution Step 1 Write the augmented matrix that represents the system. Write the augmented matrix of the system. − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 2 1 3 2 1 4 0 1 1 6 1 13 Step 2 Use row operations to obtain 1 in row 1, column 1, that is, the entry . 11 a To get 1 in row 1, column 1, interchange rows 1 and 2. [Note that this is equivalent to interchanging equations (1) and (2) of the system.] − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 2 3 1 2 4 1 0 1 1 6 13 (continued)
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