Find matrix rank depending on two parameters with entries in $\mathbb{Z/7Z}$ How do find the rank of the given matrix depending on parameters $a, b \in \mathbb{Z}_{7}$?
$$ A\in Mat_{3,4}(\mathbb{Z}_{7}), A=\begin{pmatrix} \overline{2} &b  &\overline{1}  &\overline{3} \\  \overline{-3} &\overline{3} b & \overline{2} a& \overline{0}\\   a & \overline{4} -a & \overline{-3} &\overline{1} +\overline{2} b \end{pmatrix}$$
I have found matrix rank by reducing matrix to echelon form, but there were no parameters involved in those questions, even more, two. I've spent days trying to figure out a solution by different methods (e.g by changing (-3) to 4 [they both give the same reminder when divided with 7] and then performed row operations, but it just did not worked out since i did not manage to reduce “a” to 0 etc. Then I started wondering whether I should just start by letting a/b equal to zero and watch different cases separately and reduce them to a desired form separately? I appreciate your help and explanations.

Here are some initial steps to get you started. For convenience, I'm omitting the bars.
Since we're only looking for the rank of this matrix, we can also apply column operations.
$$
\pmatrix{2 &b  &{1}  &{3} \\  
{-3} &{3} b & {2} a& {0}\\   
a & {4} -a & {-3} &{1} +{2} b} \leadsto\\
\pmatrix{1 &4b  &4  &5 \\  
{-3} &{3} b & {2} a& {0}\\   
a & {4} -a & {-3} &{1} +{2} b} \leadsto\\
\pmatrix{1 &4b  &4  &5 \\  
0 & b & {2}a-2& 1\\   
a & {4} -a & {-3} &{1} +{2} b} \leadsto\\
\pmatrix{1 &4b  &4  &5 \\  
0 & b & {2}a-2& 1\\   
0 & {4} -a-4ab & -3-4a &2 +{2} b} \leadsto\\
\pmatrix{1 &5  &4  &4b \\  
0 & 1 & {2}a-2& b\\   
0 & 2+2b & -3-4a &4-a-4ab}.\phantom{\leadsto}
$$
The first three steps were row operation, and the last was a permutation of the columns. Perhaps you can take it from here.
