The rank of a m n matrix can be judged by a number of methods, one of which is the use of the elementary row transformation of the matrix. This method is described in more detail below:
In the first step, each row of matrix A is transformed into its simplest form. The first element of each row of matrix A can be made to be either 1 or 0 by dividing each row of matrix A by the first non-zero element of that row (if any), or by subtracting each row of matrix A by the appropriate multiple of the first column of that row.
In the second step, the matrix A is transformed in elementary columns. The first element of each column of matrix A is either 1 or 0 by dividing each column of matrix A by the first non-zero element of that column (if any), or by subtracting each column of matrix A by the appropriate multiple of the first row of that column.
The third step is to count the number of 1 or 0. In the first and second steps, we transform matrix a into its simplest form, where the number of 1s or 0s in matrix a is the rank of matrix a.
It should be noted that the above method can only determine whether a m n matrix is reversible, that is, whether its rank is equal to m or n. If a m n matrix is irreversible, i.e., its rank is less than min(m,n), then the matrix does not have an inverse matrix and cannot be determinized. Therefore, before performing determinant calculations, it is necessary to determine whether the matrix is reversible.