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

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Presentation transcript:

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

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

MIT and James Orlin © To solve the system of equations: ====== x1x1 x2x2 x3x3 x4x4

MIT and James Orlin © Pivot on the element in row 1 column ====== ====== Subtract 2 times constraint 1 from constraint 2. Add constraint 1 to constraint 3. x1x1 x2x2 x3x3 x4x4

MIT and James Orlin © Pivot on the element in Row 2, Column 2 ====== Divide constraint 2 by -3. Subtract multiples of constraint 2 from constraints 1 and x1x1 x2x2 x3x3 x4x

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

MIT and James Orlin © 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 ======

MIT and James Orlin © 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