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)
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) ??