1. MIT and James Orlin © 2003 1 –Developed by James Orlin, MIT Animation of the Gauss-Jordan Elimination Algorithm

2. MIT and James Orlin © 2003 2 Solving a System of Equations To solve a system of equations, use Gauss-Jordan elimination.

3. MIT and James Orlin © 2003 3 To solve the system of equations: 1 2 2 1 1 4 2 1 2 ====== 0 6 -3 x1x1 x2x2 x3x3 x4x4

4. MIT and James Orlin © 2003 4 Pivot on the element in row 1 column 1 1 2 2 1 1 4 2 1 2 ====== ====== 0 6 -3 0 0 3 -9 6 -3 3 1 2410 6 Subtract 2 times constraint 1 from constraint 2. Add constraint 1 to constraint 3. x1x1 x2x2 x3x3 x4x4

5. MIT and James Orlin © 2003 5 Pivot on the element in Row 2, Column 2 ====== 1 0 0 2 -3 3 4 -9 6 1 -3 3 0 6 Divide constraint 2 by -3. Subtract multiples of constraint 2 from constraints 1 and 3. 113 0-2 0-30 -2 3 4 x1x1 x2x2 x3x3 x4x4 1 0 0

6. MIT and James Orlin © 2003 6 0 1 0 1 0 4 -2 3 ====== 1 0 0 3 -3 Pivot on the element in Row 3, Column 3 Divide constraint 3 by -3. Add multiples of constraint 3 to constraints 1 and 2. 10 011 0 2 x1x1 x2x2 x3x3 x4x4 Suppose x 4 = 0. What are x 1, x 2, x 3 ? 0 1 0 1 0 0

7. MIT and James Orlin © 2003 7 b1b1 b2b2 b3b3 ====== a 12 a 22 a 32 a 14 a 24 a 34 a 11 a 21 a 31 a 13 a 23 a 33 The fundamental operation: pivoting Pivot on a 23 x1x1 x2x2 x3x3 x4x4 ======

8. MIT and James Orlin © 2003 8 b1b1 b2b2 b3b3 ====== a 12 a 22 a 32 a 14 a 24 a 34 a 11 a 21 a 31 a 13 a 23 a 33 Pivot on a 23 What will be the next coefficient of b 1 ? a 32 ? of a ij for i  2? x1x1 x2x2 x3x3 x4x4 ====== a 22 /a 23 a 24 /a 23 a 21 /a 23 1 b 2 /a 23  a 11  a 11 =a 11 –a 13 (a 21 /a 23 ) 0 0