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

a_{11}= ( a_{12}(a_{41}a_{53}-a_{51}a_{43}) - a_{13}(a_{41}a_{52}-a_{51}a_{42}) ) / ( a_{42}a_{53}-a_{52}a_{43})

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