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 
 A: 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 $${[(i+1)^2-1]}^2-2i^2$$
A: This is not the smartest solution, but I want to post it as an application of inclusive-exclusive principle (I personally love this principle). 
The list/steps below may look lengthy, but it's actually very fast if you think within your head.
Plus, it is very mechanical - just follow the steps, and thus less error-prone.
We have $i+1$ integers to pick for each position.
Start point
Number of all matrix: ${(i+1)^4}$
Subtract
Number of matrix that has $0$ as its first row: ${(i+1)^2}$
Number of matrix that has $0$ as its second row: ${(i+1)^2}$
Number of matrix that has $0$ as is first column: ${(i+1)^2}$
Number of matrix that has $0$ as its second column: ${(i+1)^2}$
Plus
Number of matrix that has $0$ as its first and second row: $1$
Number of matrix that has $0$ as its first row and first column: $(i+1)$
Number of matrix that has $0$ as its first row and second column: $(i+1)$
Number of matrix that has $0$ as its second row and first column: $(i+1)$
Number of matrix that has $0$ as its second row and second column: $(i+1)$
Number of matrix that has $0$ as its first column and second column: $1$
Subtract
Number of matrix that has $0$ as its first row and second row and first column: $1$
Number of matrix that has $0$ as its first row and second row and second column: $1$
Number of matrix that has $0$ as its first row and first column and second column: $1$
Number of matrix that has $0$ as its second row and first column and second column: $1$
Plus
Number of matrix that has $0$ as its all columns and rows: $1$
Thus
The final result $=(i+1)^4-4(i+1)^2+(2+4(i+1))-4+1$
                 $$=i^2((i+2)^2-2)$$
