3.2 Gaussian Elimination

Gaussian Elimination uses two properties of equations that you learned in algebra.
  1. If both sides of an equation are multiplied by the same constant, the result is a valid equation.
  2. If two equations added together, the result is a valid equation.
We will use those properties to modify our table so that the "diagonal" entries of the table/matrix all have the value 1 and the table entries to the left and below the diagonal elements are zero. Here again is the table/matrix from the previous page:

Phase SiO2 Al2O3 K2O H2O
1 Qz 1 0 0 0
1 Sil 1 1 0 0
1 Ms 3 1.5 0.5 1
1 Water 0 0 0 1
1 Mc 3 0.5 0.5 0

Begin with the SiO2 column. Subtract multiples of the first row (the 1 Qz equation) from the rows below it so that all the SiO2 column entries below the top row are 0. Here is the result:

Phase SiO2 Al2O3 K2O H2O
1 Qz 1 0 0 0
1 Sil - 1 Qz 0 1 0 0
1 Ms - 3 Qz 0 1.5 0.5 1
1 Water 0 0 0 1
1 Mc - 3Qz 0 0.5 0.5 0

Next work on the Al2O3 column. Subtract multiples of the second row (now the
1 Sil - 1 Qz equation) from the rows below it so that all the Al2O3 column entries below the second row are 0. Here is the result:

Phase SiO2 Al2O3 K2O H2O
1 Qz 1 0 0 0
1 Sil - 1 Qz 0 1 0 0
1 Ms - 1.5 Qz - 1.5 Sil 0 0 0.5 1
1 Water 0 0 0 1
1 Mc - 2.5 Qz - 0.5 Sil 0 0 0.5 0

Next work on the K2O column. Subtract multiples of the third row (now the
1 Ms - 1.5 Qz - 1.5 Sil equation) from the rows below it so that all the K2O column entries below the third row are 0. Then multiply the third row by 2 to make the "diagonal" element 1. Here is the result:

Phase SiO2 Al2O3 K2O H2O
1 Qz 1 0 0 0
1 Sil - 1 Qz 0 1 0 0
2Ms - 3 Qz - 3 Sil 0 0 1 2
1 Water 0 0 0 1
1 Mc - 1 Qz +1 Sil - 1 Ms 0 0 0 -1

Next work on the H2O column. Subtract (or add) multiples of the fourth row (the 1 Water equation) from the rows below it so that all the H2O column entries below the fourth row are 0. Here is the result:

Phase SiO2 Al2O3 K2O H2O
1 Qz 1 0 0 0
1 Sil - 1 Qz 0 1 0 0
2Ms - 3 Qz - 3 Sil 0 0 1 2
1 Water 0 0 0 1
1 Mc - 1 Qz +1 Sil - 1 Ms +1 Water 0 0 0 0