A special class of matrices
Keith McKenzie Rogers, E. G. Straus · Pacific Journal of Mathematics · 1962
THEOREM II. If D -D^t], where t is transcendental over D u if #(A) > n > an d if A has P D , then the rows of A can be so ordered that the matrices A r of the first r rows of A have all r-by-r minors in Dand not all zero, for r = 1, 2, , n.In particular, the first row is over D, and det A eD*.If in addition we have only principal ideals, then we can reduce all but one element of the first row to zero and prove by induction: COROLLARY.If D -F[t], where #(F) > n, so K is a simple transcendental extension, then A has P D if and only if A -PTN, where P, T and N are as above.