If in any matrix of A to take any k of lines and columns and to make of elements of these lines and columns size k submatrix on k, then such submatrix is called A matrix minor. The quantity of lines and columns in the greatest such minor other than zero, is called a matrix rank.
Instruction
1. For matrixes of the small size it is possible to calculate a rank by method of search of all minors. Generally it is difficult and convenient to use method of reduction of a matrix to a triangular look. A triangular look - such kind of a matrix at which under the main diagonal of a matrix there are only zero elements. After reduction to a triangular look it is enough to count quantity of nonzero lines or columns (watching what from them it will appear less). This number will also be a matrix rank.
2. In an example the rectangular matrix of dimension 3 on 4 is considered. Already at this stage it is clear that the rank will not be higher than 3 as the smallest of dimensions is equal to 3.
3. Now it is necessary, using elementary operations, to nullify the first column of a matrix, having left nonzero only the first element in it. For this purpose increase the first line by 2 and subtract element-wise from the second line, write down result in the second line. Increase the first line by-1 and subtract from the third line to nullify the first element of the third line.
4. It was necessary to nullify the second element of the third line to receive zero elements below the main diagonal of a matrix. For this purpose subtract the second from the third line. In this case the element [3;3] matrixes also became equal to zero, it is accident, it is not necessary to try to obtain zero on the main diagonal specially. Zero lines and columns in a matrix did not appear, the rank of a matrix means it is equal to 3.