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This guide covers solving a system of three equations using substitution and matrices. It outlines the objectives including writing the augmented matrix, performing row operations, and achieving both row-echelon and reduced row-echelon forms through methods like Gaussian elimination. You'll learn how to construct the augmented matrix from a system of equations and the significance of row operations such as interchanging rows and multiplying by constants. Step-by-step examples illustrate these concepts, making it easier to solve complex linear systems.
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10.2 Systems of Linear Equations: Matrices Objectives • Write the Augmented Matrix • Write the System from the Augmented matrix • Perform Row Operations • Solve a System of Linear Equations using Matrices • Row operations • Row-echelon form (Gauss Elimination) • Reduced Row-echelon (Gauss-Jordan)
1. Augmented Matrix Given the system: The coefficient matrix is: The augmented matrix is:
1. Augmented Matrix write the augmented matrix for the warm-up
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
3. a) An example of row operations Example: Perform the following operations. (Interchange row 1 and row 2) (Add row 1 to row 2) (Add -2 times row 1 to row 2)
3 b) 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: The method for solving a system using row-echelon form is also known as Gaussian Elimination.
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
