Gaussian elimination: examples of solutions and special cases

Gaussian elimination, also called step-by-step exclusion of unknown variables, named after the great German scientist K. F. Gauss, who during his lifetime received the unofficial title of "king of mathematics". However, this method was known long before the birth of European civilization, even in the I century BC, ancient Chinese scholars used it in his writings.

Gaussian elimination is a classical method for solving systems of linear algebraic equations (SLAE). It is ideal for a quick solution to restricted size matrices.

This method consists of two moves: forward and reverse. A direct course is called consistent alignment of linear equations in triangular form, that is, zeroing of the values under the main diagonal. Reverse implies a consistent finding values of the variables, expressing each variable using the previous.

To Learn how to apply the method of Gauss easy enough knowledge of the elementary rules of multiplication, addition and subtraction of numbers.

In order to demonstrate the algorithm for the solution of linear systems by this method, let us examine one example.

So, to solve using Gaussian elimination:

X+2y+4z=3
2x+6y+11z=6
4x-2y-2z=-6

We need the second and third lines to get rid of the variable X. To do this, we add to them the first multiplied by -2 and -4 respectively. Will receive:

X+2y+4z=3
2y+3z=0
-10y-18z=-18

Now the 2nd line multiply by 5 and add it to the 3rd:

X+2y+4z=3
2y+3z=0
-3z=-18

We brought our system to a triangular form. Now undertaken to reverse. Start with the last lines:
-3z =-18
z=6.

Second line:
2y+3z=0
2y+18=0
2y=-18
y=-9

The First line:
X+2y+4z=3
X-18+24=3
X=18-24+3
X= -3

Substituting these values of the variables in the original data, convinced of the correctness of the decision.

This example can be done in a variety of any other substitutions, but the answer is supposed to be the same.

It so Happens that leading the first row contains the elements with too small values. It's not terrible, but quite complicated calculations. A solution to this problem is the Gauss method with choice of the main element in the column. Its essence consists in the following: the first line looks for the maximum in modulus element, the column in which it is set, change places with 1 m column, that is, the maximum element is the first element of the main diagonal. Next is a standard calculation process. If necessary, the procedure of swapping columns can be repeated.

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Another modified Gauss method is the method of Gauss-Jordan.

Are Used for solving square systems of linear equations, finding inverse of matrix and rank of matrix (number of nonzero rows).

The Essence of this method is that the source system by way of transformation becomes the identity matrix, with the further finding of the values of variables.

The Algorithm is as follows:

1. The system of equations is given as in the method of Gauss, in triangular form.

2. Each line is divided into a certain number so that on the main diagonal turned out unit.

3. The last line is multiplied by some number and subtracted from the next to last so as to not on the main diagonal to 0.

4. Operation 3 is repeated sequentially for all rows, until eventually not form the identity matrix.