Math "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25
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1 Math "Matrix Approach to Solving Systems" Bibiana Lopez Crafton Hills College November 2010 (CHC) 6.3 November / 25
2 Objectives: * Define a matrix and determine its order. * Write the augmented matrix for a system. * Perform elementary row operations on matrices. * Use matrices to solve a system of two equations. (CHC) 6.3 November / 25
3 Define a Matrix and Determine its Order Another way to solve systems of equations involves rectangular arrays of numbers called matrices. Definition: Matrices A matrix is any rectangular array of numbers arranged in horizontal rows and vertical columns written within brackets. Examples of matrices: 2 [ ] A = C = B = D = [ ] (CHC) 6.3 November / 25
4 Write the Augmented Matrix for a System To show how we can use matrices to solve systems of linear equations, first we let the system of equations be represented by a matrix called augmented matrix. Example 1: (Representing systems of equations using matrices) Represent each system using an augmented matrix: ( ) 2x 4y = 9 a) 5x y = 2 (CHC) 6.3 November / 25
5 Write the Augmented Matrix for a System b) a + b + c = 4 2a + 7b 2c = 0 4a + 2b 3c = 1 (CHC) 6.3 November / 25
6 Perform Elementary Row Operations on Matrices To solve a system of equations using matrices, we transform the augmented matrix into an equivalent matrix that has 1 s down its diagonal and 0 s below the 1 s. A matrix written in this form is said to be in row echelon form. Example: Non-example: (CHC) 6.3 November / 25
7 Perform Elementary Row Operations on Matrices To write an augmented matrix in row echelon form, we use three operations called elementary row operations. Elementary Row Operations Type 1: Any two rows can be interchanged. Type 2: Any row can be multiplied by a nonzero real number. Type 3: Any row can be changed by adding another row to it. (CHC) 6.3 November / 25
8 Perform Elementary Row Operations on Matrices Example 2: (Elementary row operations) Perform the following elementary row operations. a) Type 1: Interchange the rows of matrix B. B = [ ] (CHC) 6.3 November / 25
9 Perform Elementary Row Operations on Matrices b) Type 2: Multiply row 1 of matrix C by 1 2. C = [ ] (CHC) 6.3 November / 25
10 Perform Elementary Row Operations on Matrices c) Type 3: To the numbers in row 1 of matrix A, add the results of multiplying each number in row 2 by -2. A = [ ] (CHC) 6.3 November / 25
11 Use Matrices to Solve a System of Two Equations Solving Systems of Linear Equations Using Matrices: 1. Write an augmented matrix for the system. 2. Use elementary row operations to transform the matrix to row echelon form. 3. When step 2 is complete, write the resulting system and find the solution. 4. Check solution. (CHC) 6.3 November / 25
12 Use Matrices to Solve a System of Two Equations Example 3: (Solving systems of equations using matrixes) Use matrices to solve the following systems. ( ) 2x + 3y = 25 a) 3x 5y = 29 (CHC) 6.3 November / 25
13 Use Matrices to Solve a System of Two Equations (CHC) 6.3 November / 25
14 Use Matrices to Solve a System of Two Equations b) 4x + 3y z = 0 3x + 2y + 5z = 6 5x y 3z = 3 (CHC) 6.3 November / 25
15 Use Matrices to Solve a System of Two Equations (CHC) 6.3 November / 25
16 Use Matrices to Solve a System of Two Equations c) x + 2y 3z = 15 2x 3y + z = 15 4x + 9y 4z = 49 (CHC) 6.3 November / 25
17 Use Matrices to Solve a System of Two Equations (CHC) 6.3 November / 25
18 Reduced Echelon Form The final matrices of Example 3c are said to be in reduced echelon form. In general, a matrix is in reduced echelon form if the following conditions are satisfied: 1. As we read from left to right, the first nonzero entry of each row is In the column containing the leftmost 1 of a row, all the remaining entries are zeros. 3. The leftmost 1 of any row is to the right of the leftmost 1 of the preceding row. 4. Rows containing only zeros are below all the rows containing nonzero entries. (CHC) 6.3 November / 25
19 Reduced Echelon Form Example: Non-example: (CHC) 6.3 November / 25
20 Reduced Echelon Form Example 4: (Solving system of equations using matrices) Use matrices to solve the following systems. x + 3y 4z = 13 2x + 7y 3z = 11 2x y + 2z = 8 (CHC) 6.3 November / 25
21 Reduced Echelon Form (CHC) 6.3 November / 25
22 Identifying Consistent and Inconsistent Systems Example 5: (A consistent system with infinitely many solutions & an inconsistent system) Use matrices to solve the following systems. a) 3x y 2z = 1 4x + 2y + z = 5 6x 2y 4z = 9 (CHC) 6.3 November / 25
23 Identifying Consistent and Inconsistent Systems (CHC) 6.3 November / 25
24 Identifying Consistent and Inconsistent Systems b) x y + 2z = 1 3x + 4y z = 4 x + 2y + 3z = 6 (CHC) 6.3 November / 25
25 Identifying Consistent and Inconsistent Systems (CHC) 6.3 November / 25
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