  The usual Cauchy matrix approach starts from a known plain wave factor vector
$r$ and known dressed Cauchy matrix $M$. In this paper we start from a matrix
equation set with undetermined $r$ and $M$. From the starting equation set we
can build shift relations for some defined scalar functions and then derive
lattice equations. The starting matrix equation set admits more choices for $r$
and $M$ and in the paper we give explicit formulae for all possible $r$ and
$M$. As applications, we get more solutions than usual multi-soliton solutions
for many lattice equations including the lattice potential KdV equation, the
lattice potential modified KdV equation, the lattice Schwarzian KdV equation,
NQC equation and some lattice equations in ABS list.
