The matrix is the system of the elements located in the form of the rectangular table. For definition of a rank of a matrix, finding of its determinant and the return matrix it is necessary to lead the set matrix to a step look. Matrixes of a step look are also convenient for performance of other operations over matrixes.
Instruction
1. The matrix is called step if the following conditions are satisfied: • after a zero line there are only zero lines; • the first nonzero element in each next line is located more to the right, than in previous. In linear algebra there is a theorem according to which any matrix can be brought to a step look by the following elementary transformations: • change by places of two lines of a matrix; • addition to one line of a matrix of other its line increased by number.
2. Let's consider reduction of a matrix to a step look on the example of a matrix of A presented in the drawing. Solving a problem, first of all attentively study lines of a matrix. Whether it is possible to rearrange places of a line so that further it was more convenient to carry out calculations. In our case we see that it will be convenient to trade places the first and second lines. First, if the first element of the first line is equal to number 1, then it considerably simplifies the subsequent elementary transformations. Secondly, the second line will already correspond to a step look, i.e. its first element is equal to 0.
3. Further nullify all first elements of columns (except the first line). In our case it is simpler to make it since the first line begins with number 1. Therefore we consistently multiply the first line by the corresponding number and we subtract a matrix line from the turned-out line. Nullifying the third line, we multiply the first line by number 5 and it is deductible from result the third line. Nullifying the fourth line, we multiply the first line by number 2 and it is deductible from result the fourth line.
4. The next stage nullify the second elements of lines, since third line. For our example for zeroing of the second element of the third line it is enough to increase the second line by number 6 and to subtract the third line from result. For receiving zero in the fourth line it is necessary to execute more difficult transformation. It is necessary to increase the second line by number 7, and the fourth line on number 3. Thus we will receive number 21 on the place of the second element of lines. Further we subtract one line from another and we receive on the place of the second element 0.
5. And at last, we nullify the third element of the fourth line. For this purpose it is necessary to increase the third line by number 5, and the fourth line on number 3. We subtract one line from another and we receive A matrix brought to a step look.