Section 3.2 Solving Systems of Linear Equations Using Matrices

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1 Section. Solving Sstems of Linear Equations Using Matrices In Section 1. we solved X sstems of linear equations using either the substitution or elimination method. If the sstem is larger than a X, using these methods becomes tedious. In this section we ll learn how matrices can be used to represent sstem of linear equations and how to solve them, no matter the sie. In order to solve sstems of linear equation using matrices, we ll onl need the augmented matri. In a later section we ll need the coefficient and constant matrices. The following row operations, that are a result of the elimination method in Section 1., will allow us to write a linear sstem in a simplified and equivalent form. Equivalent sstems have the same solution sets. Row Operations If an of the following row operations are performed on an augmented matri, the resulting matri is an equivalent matri. Swap two rows. Notation: R1 R means Row 1 was swapped with Row. A row is multiplied b a nonero constant. Notation: 5R1 means 5 is multiplied to Row 1. A row is multiplied b a nonero constant then added to another row. Notation: R1+ R means is multiplied to Row 1 then added to Row We ll use row operations to write the augmented matri in a specific form called the row reduced form, which will allow us to read off the solution to the sstem quite easil. Section. Solving Sstems of Linear Equations Using Matrices 1

2 Row Reduced Form A matri is in row reduced form if the following conditions are satisfied. 1. If a row contains all eros, it must lie at the bottom of the matri.. The first nonero element in each row must be a one, called a leading one. Appling an row operations to obtain a leading one is called pivoting the matri about that element that becomes a one.. All other elements in each column containing a leading one are eros. This defines a unit column. 4. In an two successive rows, the leading one in the row below lies to the right of the leading one in the row above. Eample 1: Determine which of the following matrices are in row-reduced form. If a matri is not in row-reduced form, state which condition(s) is/are violated a. b. c RRF? Yes or No RRF? Yes or No RRF? Yes or No Condition(s): 1,,, 4 Condition(s): 1,,, 4 Condition(s): 1,,, 4 d e f RRF? Yes or No RRF? Yes or No RRF? Yes or No Condition(s): 1,,, 4 Condition(s): 1,,, 4 Condition(s): 1,,, g h. 0 4 i RRF? Yes or No RRF? Yes or No RRF? Yes or No Condition(s): 1,,, 4 Condition(s): 1,,, 4 Condition(s): 1,,, j RRF? Yes or No Condition(s): 1,,, 4 Section. Solving Sstems of Linear Equations Using Matrices

3 Now that we know the row reduced form, let s show how easil the solution can be read from the row reduced augmented matri. Recall that a linear sstem of equation can have one solution, no solution or infinitel man solutions. A Unique Solution Eample : The following augmented matri is in row reduced form Give the solution set for the associated linear sstem. No Solution Eample : The following augmented matri is in row reduced form Give the solution set for the associated linear sstem. Infinitel Man Solutions Eample 4: The following augmented matri is in row reduced form Give the solution set for the associated linear sstem. Section. Solving Sstems of Linear Equations Using Matrices

4 Our objective for the rest of this section will be to write augmented matrices in row reduced form. We will use the Gauss-Jordan Elimination Method to do this. Gauss-Jordan Elimination Method Basicall, ou will appl row operations to write the augmented matri in row reduced form and read off the solution(s) easil. 1. Write the augmented matri associated with the given sstem.. Use row operations to write the augmented matri in row reduced form. If at an point a row in the matri contains eros to the left of the vertical line and a nonero number to its right, stop the process the problem has no solution.. Read off the solution(s). The row operations used in Step are not unique; however, the final answer(s) will be equivalent. Eample 5: Solve the sstem of linear equations using the Gauss-Jordan elimination method. + = 1 + = 1 Section. Solving Sstems of Linear Equations Using Matrices 4

5 Eample 6: Solve the sstem of linear equations using the Gauss-Jordan elimination method. 8 = 9 + = 7 5 = 1 Section. Solving Sstems of Linear Equations Using Matrices 5

6 Eample 7: Solve the sstem of linear equations using the Gauss-Jordan elimination method. = 7 14 = 14 6 = 6 Section. Solving Sstems of Linear Equations Using Matrices 6

7 Eample 8: Solve the sstem of linear equations using the Gauss-Jordan elimination method. + = = 6 = 1 Section. Solving Sstems of Linear Equations Using Matrices 7

8 Section. Solving Sstems of Linear Equations Using Matrices 8 Eample 9: Solve the sstem of linear equations using the Gauss-Jordan elimination method. 5 1 = + = = +

9 Eample 10: Solve the sstem of linear equations using the Gauss-Jordan elimination method. + = + = = Section. Solving Sstems of Linear Equations Using Matrices 9

10 Section. Solving Sstems of Linear Equations Using Matrices 10 Tr this one: Solve the sstem of linear equations using the Gauss-Jordan elimination method = + = + = +