Rank and rational functions

Suppose a matrix A (say size 6x6) has rank 2, then any diagonal element can be expressed as rational functions of certain off-diagonal elements. (Using certain 3x3 matrix determinants which are 0)

For example
a₁₁ = ( a₁₂(a₄₁a₅₃-a₅₁a₄₃) - a₁₃(a₄₁a₅₂-a₅₁a₄₂) ) / ( a₄₂a₅₃-a₅₂a₄₃ )

Suppose it has rank 3, can diagonal elements be expressed as rational functions of any set of off-diagonal elements (only off-diagonal) ??

Hey srkri.

I'm not familiar with rational functions but wikipedia gives a ratio of two polynomials (which I'm sure is not what you are referring to), so could you please explain what a rational function is in your context?

I do mean rational functions as ratio of two polynomials.