The rank of a matrix is the largest number of rows and columns in a minor that is not equal to zero. Determination of the rank of a matrix is carried out in different ways, the most convenient and simple is to bring it to a triangular form.
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To determine the rank of a small matrix, use an enumeration of all minors or, which is much simpler, reduce the matrix to a triangular form. In this case, only zero elements are located under its main diagonal. The rank of the matrix in this case is determined by the number of their rows or columns.
If their number is different, use the smallest value, that is, it cannot be greater or less than the smallest number of zero elements. This method of calculating a matrix is quite convenient, in contrast to enumeration of minors, since the calculations are much easier, and the result will be the same.
Zero the first column of the matrix, but note that the very first element must be left unchanged. To do this, multiply the first row of the matrix by 2 and subtract from the second row element by element. Write down the result of calculations you received in the second line, then multiply the first by minus one and subtract from the third, thereby zeroing the first element contained in the third line.
Go to the last step - zeroing out the second element contained in the third row of the matrix whose rank you want to determine. After that, you get zero elements that are lower than the main diagonal. Subtract the second from the third row of the matrix, if the element of the matrix becomes equal to zero, most likely it will not be on purpose, so there is no need to specially bring the matrix to zero values on its main diagonal.
Determine the rank of the matrix according to the number of zero elements. If a situation arises when one of the sides has more than zero values, use the other side of the triangular matrix with the smallest number of them, otherwise its rank will be determined incorrectly.