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

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

3. To solve the system of equations:x1 x2 x3 x4 1 2 4 1 = 2 1 -1 -1 6 -1 1 2 2 -3
MIT and James Orlin © 2003

4. Pivot on the element in row 1 column 1x1 x2 x3 x4 1 1 2 2 4 1 4 1 = = 2 -3 1 -9 -1 -1 -3 6 6 -1 1 3 6 2 2 3 -3 -3
Subtract 2 times constraint 1 from constraint 2. Add constraint 1 to constraint 3.
MIT and James Orlin © 2003

5. Pivot on the element in Row 2, Column 2x1 x2 x3 x4 1 1 2 -2 4 -1 1 4 = 1 -3 3 -9 -3 1 6 -2 3 -3 6 3 -3 3
Divide constraint 2 by -3. Subtract multiples of constraint 2 from constraints 1 and 3.
MIT and James Orlin © 2003

6. Pivot on the element in Row 3, Column 3x1 x2 x3 x4 1 1 -2 -1 -1 2 4 = 1 1 3 1 1 -2 1 -3 1 -1 3
Divide constraint 3 by -3. Add multiples of constraint 3 to constraints 1 and 2.
MIT and James Orlin © 2003
Suppose x4 = 0. What are x1, x2, x3?

7. The fundamental operation: pivotingx1 x2 x3 x4
a11
a12
a13
a14 b1 = =
a21
a22
a23
a24 b2
a31
a32
a33
a34 b3
Pivot on a23
MIT and James Orlin © 2003

8. What will be the next coefficient of b1? a32? of aij for i 2?
Pivot on a23
a11 =a11 –a13(a21/a23) x1 x2 x3 x4
a11
a11
a12
a13
a14 b1 = =
a21/a23
a21
a22/a23
a22 1
a23
a24
a24/a23
b2/a23 b2
a31
a32
a33
a34 b3
What will be the next coefficient of b1? a32? of aij for i 2?
MIT and James Orlin © 2003