Warm up Find the dimensions of the following matrices:Find the dimensions of the following matrices: For the first matrix find a 213. For the first matrix find a 21

Gauss-Jordan Elimination Objective: To solve system of equations using Gauss-Jordan elimination of an augmented matrix.

3 Augmented Matrix for a System of Equations Given a system of equations we can talk about its coefficient matrix and its augmented matrix.Given a system of equations we can talk about its coefficient matrix and its augmented matrix. To solve the system we can now use row operations instead of equation operations to put the augmented matrix in row echelon form.To solve the system we can now use row operations instead of equation operations to put the augmented matrix in row echelon form.

4 Row-Echelon Form A matrix is in row-echelon form if:A matrix is in row-echelon form if: The lower left quadrant of the matrix has all zero entries.The lower left quadrant of the matrix has all zero entries. In each row that is not all zeros the first entry is a 1.In each row that is not all zeros the first entry is a 1. The diagonal elements of the coefficient matrix are all 1The diagonal elements of the coefficient matrix are all 1

Gauss-Jordan Elimination Solve:Solve: Only care about numbers – form “tableau” or “augmented matrix”:Only care about numbers – form “tableau” or “augmented matrix”:

Gauss-Jordan Elimination Given:Given: Goal: reduce this to trivial system and read off answer from right columnGoal: reduce this to trivial system and read off answer from right column

Gauss-Jordan Elimination Basic operation 1: replace any row by linear combination with any other rowBasic operation 1: replace any row by linear combination with any other row Here, replace row1 with 1 / 2 * row1 + 0 * row2Here, replace row1 with 1 / 2 * row1 + 0 * row2

Gauss-Jordan Elimination Replace row2 with row2 – 4 * row1Replace row2 with row2 – 4 * row1 Negate row2Negate row2

Gauss-Jordan Elimination Replace row1 with row1 – 3 / 2 * row2Replace row1 with row1 – 3 / 2 * row2 Read off solution: x= 2, y = 1Read off solution: x= 2, y = 1

Gauss-Jordan Elimination For each row i:For each row i: Multiply row i by 1/a iiMultiply row i by 1/a ii For each other row j:For each other row j: Add –a ji times row i to row jAdd –a ji times row i to row j At the end, left part of matrix is identity, answer in right partAt the end, left part of matrix is identity, answer in right part

11 Gauss-Jordan Elimination In Gauss-Jordan elimination, we reduce the augmented matrix until we get a row equivalent matrix in reduced row-echelon form. (r-e form where every column with a leading 1 has rest zeros)In Gauss-Jordan elimination, we reduce the augmented matrix until we get a row equivalent matrix in reduced row-echelon form. (r-e form where every column with a leading 1 has rest zeros)

12 Gauss-Jordan Elimination Let us consider the set of linearly independent equations. Augmented matrix for the set is:

13 Gauss-Jordan Elimination Step 1: make the first x = 1. -(R1 + R2)

14 Gauss-Jordan Elimination Step 2: Eliminate the other 2 x’s from the first column. 3R 1 + R 2 R 3 -5 R 1

15 Gauss-Jordan Elimination Step 3: Create the 1 in the second row second column R 1 + R 2 R 2 /2 Step 4: Eliminate the other y’s 8R 2 – R 3

16 Gauss-Jordan Elimination Step 5: Create the 1 in the 3 rd row 3 rd column. R 3 /-168

17 Gauss-Jordan Elimination Step 6: Eliminate the other z’s. (53/2)R 3 + R 1 (29/2)R 3 + R 2 x=2 y=-3 z=4

Practice x + 2y = -2x + 2y = -2 2x +6y = 22x +6y = 2

Practice x + y + z = 4x + y + z = 4 2x – y +2z = 112x – y +2z = 11 x + 2y + 2z =6x + 2y + 2z =6

Sources e05_linsys.pptwww.cs.princeton.edu/.../cos323_s06_lectur e05_linsys.ppt