Is it possible to mathematically add zeros to rows of a matrix that aren't of the same size as the rest of the matrix? Lets say I have a some lists: 
$$a=\{1,4,4\}$$
$$b=\{8,4\}$$
$$c=\{9,8,0\}$$
and I know that each list should be of length 3. So $b$ is "corrupted". 
If I arrange these values in a certain order we have $$\text{cell} (1,1),a1 + \text{cell} (2,1),b1 = \text{cell} (3,1),c1.$$ And, if this pattern exist throughout an nxn matrix, is it possible to employ linear algebra techniques to solve for the missing cell? Is it also possible to zero in on the cell that is missing? 
That is to know exactly cell$(2,3)$ is the cell that is missing or should be zero.
What if I have multiple missing values?
I will appreciate any honest help and direction to help me get closer to an answer. Thanks community!
 A: It looks like you're asking what happens if you have a matrix of values such that (say) the columns all follow a simple arithmetic rule (such as that the first row plus the second row equals the third row) but some of the values are missing.
If you know exactly where the values are and you'd like to fill them in (like a Sudoku puzzle), you can use linear algebra. In fact, if the relation is just simple arithmetic, then you can express the row with missing elements as a linear combination of the other rows. Just find the kernel of the matrix as best you can; the kernel will tell you what the coefficients of that linear combination can be.
If you don't know where the values are (you just know that some rows are shorter than they should be), then linear algebra itself won't be of direct use. Instead, you can try expressing the problem as a constraint satisfaction problem and search for the missing values that way.
Sometimes, if too many values are missing, you won't be able to recover the cells uniquely. For example:
$$\begin{bmatrix}1 & 5 & -2 \\ x & 2 & 8 \\ y & 1 & -18\end{bmatrix}$$
(Here, the rule is that the first column minus twice the second column equals the third column. Unfortunately, there are many possible values for the deleted cells.)
