How many $2 \times 2$ matrices are there with entries from the set ${\{0,1,2,...,i}\}$ in which there are no zeros rows and no zero columns? How many $2 \times 2$ matrices are there with entries from the set ${\{0,1,2,...,i}\}$ in which there are no zeros rows and no zero columns? 
attempt:
Suppose we have a matrix $ \begin{pmatrix} 
a & b \\
c & d 
\end{pmatrix}$. Then we don't want $ \begin{pmatrix} 
0 & 0 \\
0 & 0 
\end{pmatrix}$ or $ \begin{pmatrix} 
0 & 0 \\
c & d 
\end{pmatrix}$ or $ \begin{pmatrix} 
a & b \\
0 & 0
\end{pmatrix}$.
Similarly for the columns.
Then we have the first case:
that all of the $a,b,c,d \neq 0$,and so we have $i^4$ options to choose such a matrix.
second case: we have that $ \begin{pmatrix} 
0 & b \\
c & 0 
\end{pmatrix}$ or $ \begin{pmatrix} 
a & 0 \\
0 & d 
\end{pmatrix}$. so we have $i^2$  or $i^2$ on both,so $2i^2$ ways to choose a matrix.
third case : either the $a,b,c,d$ is zero and the rest nonzero.
I am not really sure. I would add the cases to get the final answer.
Can someone please help me? Any feedback would really help.
Thanks 

You may count the number of two tuples that can be formed from the elements of your set. clearly there are $(i+1)^2$ two tuples. Now the first entry should not be the vector $(0,0)$ so you have $(i+1)^2-1$ options for first row of marix. You have number of options for second row but then you have to subtract matrices of form $$ \begin{pmatrix} 
0 & b \\
0& d
\end{pmatrix}$$ and $$ \begin{pmatrix} 
a& 0\\
c & 0 
\end{pmatrix}$$ (where $a$, $b$, $c$ and $d$ are non zero)
 which are $2i^2$ in number.
so your answer should be $${[(i+1)^2-1]}^2-2i^2$$
