Not possible. Renato is correct, in that you have essentially only 10 pieces of information, in the form of 10 linear equations.
However, you have 55 unknowns. You can pick some subset 45 of those unknowns, and set them to zero (or any value you choose) and then solve for the other 10 unknowns. Not ANY arbitrary subset of course. Thus...
The single element from row 1 is easily computed.
Next, you can choose any one element from row 2 to set to any chosen value. Then the other element is given.
Repeat for each row, setting n-1 elements on the nth row to some arbitrary values, then compute the one element that remains.
You can do no better than this though. Sorry, but mathematics is unforgiving.