where □ can be 0 or 1, there are 2 c ( $^{r}$−c ) distinct unitary row operations. Now we consider matrices in which the leading ones are shifted to the right. Let l$_{j}$ denote the shift amount of the leading 1 of j -th row from its initial position for j = 1 , · · · , c . It can be observed that l$_{j}$ ≤ l$_{i}$ if j < i . For a set { l$_{1}$, l$_{2}$, · · · , l$_{c}$ } , the number of □ is c ( r − c ) − ∑ c i =1 l$_{i}$ , and hence there are 2 c ( $^{r}$−c $^{)}$− ∑ c i $_{=1}$l$_{i}$ distinct unitary row operations. Therefore, summing over all possible sets of { i$_{1}$, · · · , i$_{c}$ } shows that there is a total of distinct unitary row operations up to Gaussian elimination. Example 3. For r = 4 and c = 2, possible matrices of M$_{V}$ are N ( r, c ) = $^{r}$−c ∑ l$_{c}$ =0 l$_{c}$ ∑ l$_{c}$$_{−}$$_{1}$ =0 l$_{c}$$_{−}$$_{1}$ ∑ l$_{c}$$_{−}$$_{2}$ · · · l$_{2}$ ∑ l$_{1}$ =0 2 c ( $^{r}$−c $^{)}$− ∑ c i $_{=1}$l$_{i}$   $^{1}$ 0 □ □ 0 1 □ □    $_{,}$  $^{1}$ □ 0 □ 0 0 1 □    $_{,}$  $^{1}$ □ □ 0 0 0 0 1    $_{,}$  $^{0}$ 1 0 □ 0 0 1 □    $_{,}$  $^{0}$ 1 □ 0 0 0 0 1    $_{,}$  $^{0}$ 0 1 0 0 0 0 1    . 2 4 + 2 3 + 2 2 + 2 2 + 2 1 + 2 0 = 35 There are distinct row operations. ✷ The function N ( r, c ) has a symmetric property as in the following lemma and some closed forms of N ( r, c ) are listed in Table 2. Lemma 4. N ( r, c ) = N ( r, r − c ) for any r and 0 ≤ c ≤ r. Proof. We prove this lemma by mathematical induction. Assume N ( l, c ) = N ( l, l − c ) holds for l = 1 , · · · , r and c = 0 , · · · , l . It suffices to assume c < r − c or 2 c < r . Consider the 24