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Warm up

Warm up. Find the dimensions of the following matrices: 1. 2. 3. For the first matrix find a 21. Gauss-Jordan Elimination. Objective: To solve system of equations using Gauss-Jordan elimination of an augmented matrix.

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Warm up

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  1. Warm up Find the dimensions of the following matrices: 1. 2. 3. For the first matrix find a21

  2. 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. • 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: • 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. • The diagonal elements of the coefficient matrix are all 1

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

  6. Gauss-Jordan Elimination • Given: • Goal: reduce this to trivial systemand read off answer from right column

  7. Gauss-Jordan Elimination • Basic operation 1: replace any row bylinear combination with any other row • Here, replace row1 with 1/2 * row1 + 0 * row2

  8. Gauss-Jordan Elimination • Replace row2 with row2 – 4 * row1 • Negate row2

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

  10. Gauss-Jordan Elimination • For each row i: • Multiply row i by 1/aii • For each other row j: • Add –aji times row i to row j • At 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)

  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. 3R1 + R2 R3 -5 R1

  15. Gauss-Jordan Elimination Step 3: Create the 1 in the second row second column R2/2 Step 4: Eliminate the other y’s R1 + R2 8R2 – R3

  16. Gauss-Jordan Elimination Step 5: Create the 1 in the 3rd row 3rd column. R3/-168

  17. Gauss-Jordan Elimination Step 6: Eliminate the other z’s. (53/2)R3 + R1 (29/2)R3 + R2 x=2 y=-3 z=4

  18. Practice • x + 2y = -2 • 2x +6y = 2

  19. Practice • x + y + z = 4 • 2x – y +2z = 11 • x + 2y + 2z =6

  20. Sources • www.cs.princeton.edu/.../cos323_s06_lecture05_linsys.ppt • www.imanighana.com/Lecture12S.ppt

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