Perhaps something like this?

m = Table[If[i == j, 0, RandomInteger[{0, 5}]], {i, 10}, {j, 10}];
Do[If[i == 3, m[[i, i]] = 0, m[[3, i]] == RandomInteger[{3, 5}]], {i, 
  4, 10}]
m[[3, 10]] = 30 - Sum[m[[3, j]], {j, 9}];
m // MatrixForm
sum1 = Sum[m[[j, 1]], {j, 10}]
sum2 = Sum[m[[3, j]], {j, 10}]

FIrst, your using x not as a matrix but as a List[], as opposed to Array[] Function[] or Expression[]. Mathematica does not use the word "matrix" to describe it's data structure(except inside a linear algebra pack and other special adaptations).

There are many syntaxes - and some new ones. But to answer the question I think most people will need to know this:

"to apply these constraints in the Minimize function"

To apply the constrains in which Minimization function? You cannot apply conditions inside the built-in function because "Evaluate" doesn't work in that way or order.

You can check constraints and correct constraints using code or functions, have them handled as a function or act inside an array as expressions or as expressions, and there are more possibilities.

"X(i,j) < some condition> where i=1,2,... n"

Is something you would see in a low level programming class text book source comment (you will find easier and better ways to do it in Mathematica). Back to Minimize: it's not just that you want conditions - how to do them really depends on what are you trying to do.

Being homework - you may need to study more about Mathematica before using it in a timed project (as in homework due tomorrow). Mathematica can be far easier and far more powerful than other languages but it is not a small language so it is not simply encapsulated in a few phrases like some small code languages are.