The matrix is called a step if the following conditions are met:
• after the zero rows are only zero rows;
• the first nonzero entry in each subsequent row is located more to the right than the previous.
In linear algebra there is a theorem that any matrix can be reduced to a speed meant by the following elementary transformations:
• changing places two rows of the matrix;
in addition to one matrix row to another row multiplied by a number.
Consider the reduction of a matrix to speed mind the example of the matrix A, is shown in Fig. Solving the problem, first carefully read row of the matrix. Is it possible to rearrange the lines so that in future easier to calculate. In our case, we see that it is convenient to swap the rst and second lines. First, if the first element of the first line is number 1, then this greatly simplifies subsequent elementary transformations. Secondly, the second line will have to match the speed of sight, i.e., its first element is equal to 0.
Next, zero out all the elements of the first column (except first row). In our case it is easier to do because the first row starts with number 1. Therefore, we sequentially multiply the first line by the corresponding number and subtract from the row line of the matrix. Resetting the third row, multiply the first line by the number 5 and subtract the result from the third row. Zeroing the fourth row, multiply the first row by the number 2 and subtract from result the fourth line.
The next step will zero out the second row elements, starting from the third row. For our example, to zero out the second element of the third line, it is sufficient to multiply the second line by the number 6, and subtract from the result the third place. To obtain a zero in the fourth line will have to perform more complex transformation. It is necessary to multiply the second line by the number 7, and the fourth place to the number 3. Thus we get in place of the second element row number 21. Then subtract one line from another and get in place the second element is 0.
Finally, reset the third element of the fourth row. To do this, multiply the third row by the number 5, and the fourth row by the number 3. Subtract one row from another and the resulting matrix A given by step mind.