Suppose that there is a $n \times n$ matrix. Then there will be entries $A_{ij}$ where i and j represent row and column.
Equations contain entries of a matrix, $A_{ij}$ and when the equations are solved, we will get each entry. (Add: By equations I mean like ${A_{11}}^2 + {A_{12}}^3+..$ where $A_{ij}$ would be entries of a matrix.)
Is there any way to represent the matrix using a system of only $kn$ equations where $k$ is constant for all combinations of integer entries? (Obviously, linear equations will not do this.)
Any variation of the matrix is allowed if the variation respect the following: entries ($n \times n$ of them) in the same column and row must be in the same column and row.
Edit: In linear algebra, for example, if there are $n \times n$ distinct equations, then you will be able to get entries of a matrix. This is a-bit different question.