3. Replace any row by the sum of that row and a constant multiple of any other row.

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1 Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or dimension m n. The real numbers that make up the matrix are called entries or elements of the matrix. The entry in the ith row and jth column is denoted by a A matrix with only one column or one row is called a column matrix (or column vector) or row matrix (or row vector), respectively. ij Example : Given A 7 7 9, a. what is the dimension of A? b. identify a 4. c. identify a. Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

2 Systems of Linear Equations in Matrix Form In order to write a system of linear equations in matrix form, first make sure the like variables occur in the same column. Then we ll leave out the variables of the system and simply use the coefficients and constants to write the matrix form. Given the following system of equations: x + 4y + 6z = x + 8y + z = 7 x + y + z = The coefficient matrix is: The constant matrix is: ( 7) 4 6 The augmented matrix is: 8 7 Example : Give the coefficient, constant and augmented matrix for the system of equations. x 4y = y + z = 9 x + y z = 8 Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

3 Section. Solutions of Linear Systems by the Gauss-Jordan Method Page As you may recall from College Algebra, you can solve a system of linear equations in two variables easily by applying the substitution or addition method. Since these methods become tedious when solving a large system of equations, a suitable technique for solving such systems of linear equations will consist of Row Operations. The sequence of operations on a system of linear equations are referred to equivalent systems, which have the same solution set. Row Operations. Interchange any two rows. R R. Multiply any row by a nonzero constant R R. Replace any row by the sum of that row and a constant multiple of any other row. R R R 7 7 Reduced Row Echelon Form(RREF) An m n augmented matrix is in row-reduced form if it satisfies the following conditions:. Each row consisting entirely of zeros lies below any other row having nonzero entries. the correct row-reduced form. The first nonzero entry in each row is (called a leading ). the correct row-reduced form

4 Section. Solutions of Linear Systems by the Gauss-Jordan Method Page 4. If a column contains a leading, then the other entries in that column are zeros. the correct row-reduced form 4. In any two successive (nonzero) rows, the leading in the lower row lies to the right of the leading in the upper row. the correct row-reduced form Example : Determine which of the following matrices are in row-reduced form. If a matrix is not in row-reduced form, state which condition is violated. a. d. 9 b. e. 6 4 c. f.

5 The Gauss-Jordan Elimination Method. Write the augmented matrix corresponding to the linear system.. Use row operations to write the augmented matrix in row reduced form. If at any point a row in the matrix contains zeros to the left of the vertical line and a nonzero number to its right, stop the process, as the problem has no solution.. Read off the solution(s). There are three types of possibilities after doing this process. Unique Solution Example 4: The following augmented matrix in row-reduced form is equivalent to the augmented matrix of a certain system of linear equations. Use this result to solve the system of equations. (a) ( 4 ) (b) ( 7) Example : Solve the system of linear equations using the Gauss-Jordan elimination method. x y x y Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

6 Example 6: Solve the system of linear equations using the Gauss-Jordan elimination method. x + y = + z x + z = 6 + y x + y + z = Section. Solutions of Linear Systems by the Gauss-Jordan Method Page 6

7 Section. Solutions of Linear Systems by the Gauss-Jordan Method Page 7 Infinite Number of Solutions Example 7: The following augmented matrix in row-reduced form is equivalent to the augmented matrix of a certain system of linear equations. Use this result to solve the system of equations. Example 8: Solve the system of linear equations using the Gauss-Jordan elimination method. z y x z y x z y x

8 Example 9: Solve the system of linear equations using the Gauss-Jordan elimination method. x + y + z w = 4 x + y + w = x + y + z = Section. Solutions of Linear Systems by the Gauss-Jordan Method Page 8

9 A System of Equations That Has No Solution In using the Gauss-Jordan elimination method the following equivalent matrix was obtained (note this matrix is not in row-reduced form, let s see why): 4 4 Look at the last row. It reads: x + y + z = -, in other words, = -!!! This is never true. So the system is inconsistent and has no solution. Systems with No Solution If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line, then the system of equations has no solution. Example : Solve the system of linear equations using the Gauss-Jordan elimination method. x y = x + y = x 4y = Section. Solutions of Linear Systems by the Gauss-Jordan Method Page 9

10 Example : Solve the system of linear equations using the Gauss-Jordan elimination method. x + y 4z = 4x y + 6z = 6 Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

11 Example : A convenience store sells sodas one summer afternoon in -, 6-, and -oz cups (small, medium, and large). The total volume of soda sold was 76 oz. (a) Suppose that the prices for a small, medium, and large soda are $, $., and $.4, respectively, and that the total sales were $8.4. How many of each size did the store sell? Let x = the number of small soda; y = the number of medium soda; z = the number of large soda. Soda Sales Small Medium Large Total Number Weight Price Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

12 (b) Suppose the prices for small, medium, and large sodas are changed to $, $, and $, respectively, but all other information is kept the same. How many of each size did the store sell? Soda Sales Small Medium Large Total Number Weight Price Section. Solutions of Linear Systems by the Gauss-Jordan Method Page

13 (c) Suppose the prices are the same as in part (b), but the total revenue is $48. Now how many of each size did the store sell? Soda Sales Small Medium Large Total Number Weight Price (d) Give the solutions from part (c) that have the smallest and largest numbers of large sodas. Section. Solutions of Linear Systems by the Gauss-Jordan Method Page