1. 10.2 Systems of Linear Equations: Matrices Objectives Objectives 1.Write the Augmented Matrix 2.Write the System from the Augmented matrix 3.Perform Row Operations 4.Solve a System of Linear Equations using Matrices a)Row operations b)Row-echelon form (Gauss Elimination) c)Reduced Row-echelon (Gauss-Jordan)

2. 1. Augmented Matrix Given the system: The coefficient matrix is: The augmented matrix is:

3. 1. Augmented Matrix write the augmented matrix

4. 2. Write System from augmented matrix 1)Write the system 2)Now solve it ! #1

5. 2. Write System from augmented matrix Write the system and solve. #2

6. 3. Row Operations Notation: new row after row operations are applied original row Multiply row i by a constant k Interchange row 1 and row 2

7. Perform each operation (using the “previous” matrix for each) 1. (Interchange row 1 and row 2) 2. (Add row 3 to row 2) 3. (Add 3 times row 1 to row 3) 4. 3. a) An example of row operations 5. What operation would put 1 in position 2,2 ?

8. 3 b) Row-Echelon Form row-echelon form Augmented matrix reduced to a form with 1’s on diagonal and 0’s beneath diagonal is called row-echelon form A 2x2 system would have the form: A 3x3 system would have the form: Gaussian Elimination. The method for solving a system using row-echelon form is also known as Gaussian Elimination.

9. Note: The augmented matrix from warm-up is in row-echelon form Example 1: Solve the 2x2 system p. 755 #37 Handout 10.2: Solving Linear Systems using Matrices Note: The augmented matrix from warm-up is in row-echelon form Example 1: Solve the 2x2 system p. 755 #37 Handout 10.2: Solving Linear Systems using Matrices

10. Review: Solve a system of 3 equations using Substitution