If (c_i1,c_i2· · · c_im) is a nonzero row of a matrix (c_ij), then its leading entry is c_it where r is the first integer such that c_it ≠ 0. A matrix C = (c_ij) over a division ring D is said to be in reduced row echelon form provided: 

(i) For some r ≥ 0 the first r rows of C are nonzero (row vectors) and all other rows are zero; 

(ii) The leading entry of each nonzero row is 1_D; (iii) if c_ij = 1_D is the leading entry of row i, then C_kj = 0 for all k ≠ i; (iv) if c_1ji,C_2ji, ... , c_rjr, are the leading entries of rows 1,2, ... , r, then j₁ < j₂ < · · · < j_r.

(a) If C is in reduced row echelon form, then rank C is the number of nonzero rows.

(b) If A is any matrix over D, then A may be changed to a matrix in reduced row echelon form by a finite sequence of elementary row operations.