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Gazprom and the European Union


To perform Gaussian elimination starting with the system of equations


compose the "augmented matrix equation"


Here, the column vector in the variables is carried along for labeling the matrix rows. Now, perform elementary row operations to put the augmented matrix into the upper triangular form


Solve the equation of the th row for , then substitute back into the equation of the st row to obtain a solution for , etc., according to the formula


In Mathematica, RowReduce

performs a version of Gaussian elimination, with the equation being solved by GaussianElimination[m_?MatrixQ, v_?VectorQ] := Last /@ RowReduce[Flatten /@ Transpose[{m, v}]]

LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.

A matrix that has undergone Gaussian elimination is said to be in echelon form.

For example, consider the matrix equation


In augmented form, this becomes


Switching the first and third rows (without switching the elements in the right-hand column vector) gives


Subtracting 9 times the first row from the third row gives


Subtracting 4 times the first row from the second row gives


Finally, adding times the second row to the third row gives


Restoring the transformed matrix equation gives


which can be solved immediately to give , back-substituting to obtain (which actually follows trivially in this example), and then again back-substituting to find

Gazprom and the European Union

Date: 2015-01-02; view: 662

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