1. Matrices Write and Augmented Matrix of a system of Linear Equations Write the system from the augmented matrix Solve Systems of Linear Equations using matrices

2. Matrix A matrix is defined as a rectangular array of numbers Col 1 Col 2Col n Row 1 Row 2 Row i

3. Continued Each number of the matrix has two indexes: the row index i and the column index j. We will use matrix notation to represent a system of equations. The matrices used to represent systems of linear equations are called augmented matrices.

4. Writing the Augmented Matrix Write the augmented matrix of a system of equations 2x - y+ z=0 a.3x -4y = -6b. x +z -1 = 0 2x -3y = -5x + 2y -8 = 0

5. Row Operations on a Matrix Row operations on a matrix are used to solve systems of equations when the system is written as an augmented matrix. 1. Interchange any two rows 2. Replace a row by a non zero constant multiple of that row 3. Replace a row by the sum of that row and a nonzero constant multiple of some other row

6. Solving a system of Linear Equation using Matrices and Row Echelon Form A matrix is in row echelon form when 1.The entry in row 1 column 1 is 1 and 0’s appear below it 2.The first non zero entry in each row after the first row is a 1 and 0’s appear below it, and it appears to the right of the first non zero entry in any row above it 3.Any rows that contain all 0’s to the left of the vertical bar appear at the bottom

7. Advantages 1.The process is algorithmic; that is it consists of repetitive steps that can be programmed on a computer 2.The process works on any system of linear equations; no matter how many equations or variables are present

8. Solving Systems of Equations Solve:

9. Matrix Method for Solving a System 1.Write the augmented matrix that represents the system 2.Perform row operations that place the entry 1 in row 1 column 1 3.Perform row operations that leave the entry 1 in row 1 column 1 unchanged, while causing 0’s to appear below column 1 4.Perform row operations that place the entry 1 in row 2 column 2 but leave the entries in columns to the left unchanged.

10. Continued 5.Now repeat Step 4 placing a 1 in the next row, but one column to the right. Continue until the bottom row or the vertical bar is reached 6.The matrix that results is the row echelon form of the augmented matrix. Analyze the system of equations corresponding to it to solve the original system.

11. Reduced Row Echelon form Sometimes a better form is reduced row echelon form. In this form row operations are used to obtain entries that are 0 above as well as below the leading 1 in a row.

12. Matrix Method for solving system We are going to use our calculator!!!!!!! Look on page 706 if you want to see the detailed method