Duality relations connected with Gaussian elimination

Using the duality relations associated with Gauss's method it is shown that: 1. Any main submatrix of a totally non-degenerate matrix $A$; is stipulated (in the sense of an arbitrary monotonic norm) to be not inferior to $A$; itself; 2. If the inverse of matrix $A$; is diagonally dominant with respect to the columns, then the linear system $Ax=b$ is regularly solved by Jordan's method without the main element being chosen.