Is conditional arrangement of cells in a mxn matrix unique?

[Adel Makram]

How many ways to arrange cells of k possible values in a mxn matrix provided that sums of all rows and columns are known?
For example, if we have a 5x3 matrix and 10 possible values ( from 0 to 9) that can be assigned for each cell, then how many ways to arrange cells in that matrix satisfying the given sums of rows and columns? Will it be a unique arrangement? And will the answer be affected by the values of m, n and k no matter how big k will be for example?

 

[Adel Makram]

I tried 2x2 matrix and I found no unique solutions to it. I fact, I tried 2xn matrices and I found no unique solutions to them ( this is probably because all 2xn matrices are still of rank 2).

But I tried few 3x3 matrices and I found that it is not easy to find a non-unique solution. This is because in all cases I have to make more than 2 random selections of values.

 

[Stephen Tashi]

How many ways to arrange cells of k possible values in a mxn matrix provided that sums of all rows and columns are known?

You need to state the problem clearly. What does "k possible values mean"?

Do you mean "k distinct values, some which may be repeated?" Do you mean "k non-zero values, some of which may be identical"? Are the k values given or can they be assigned arbitrarily?

 

[Adel Makram]

You need to state the problem clearly. What does "k possible values mean"?

Do you mean "k distinct values, some which may be repeated?" Do you mean "k non-zero values, some of which may be identical"? Are the k values given or can they be assigned arbitrarily?

Yes "k values" means k distinct but arbitrary values which can be repeated. For example a set of integers from zero to 9.

 

[HallsofIvy]

By the way, you are talking about "arrays", not "matrices".

 

[Stephen Tashi]

Yes "k values" means k distinct but arbitrary values which can be repeated. For example a set of integers from zero to 9.

It's not clear to me what you mean. To pose the problem, do we tell the problem solver the specific k values that must be used (such as saying they must be the set of integers from zero to 9)? Or do we allow the problem solver to pick any k distinct values that he wants to?

Either way, without considering the specific sums that are given for the rows and columns, I don't see that there is a general rule about whether a solution exists.

 

[Adel Makram]

It's not clear to me what you mean. To pose the problem, do we tell the problem solver the specific k values that must be used (such as saying they must be the set of integers from zero to 9)? Or do we allow the problem solver to pick any k distinct values that he wants to?.

The specific sums of all rows and columns is strict and must be satisfied. This is the only information available about the matrix. The solver is free to use any numbers from the set of k-values with or without repetition as he wishes as long as he satisfies the aforementioned sums.