Building other matrices from these ones

Question

First of all, I want to know if it is possible to make a constraint in a set of matrices that I have to obtain a subset. For example, if I have a set of square matrices A, can I build a set B formed by the elements of A that has no null determinant?

If the answer is yes, I would like do do the following.

With

m = Table[Table[1, {4}], {3}]
S = ReplacePart[m, #] & /@ Thread[Rest[Subsets@Position[m, 1]] -> -1]
MatrixForm /@ S
Length@S

We obtain from

m={{1, 1, 1, 1}, {1, 1, 1, 1}, {1, 1, 1, 1}}

the set of all matrices like m but changing the 1's to -1's. Now I want to add the line {1,1,1,1} to each matrix of the set S and obtain a set S1. Then I want to know how can I obtain the subset M of S1 formed by all matrices that have no null determinant.

The final effect of what I am trying to achieve is to obtain the set of all 4 x 4 invertible matrices with entries 1 or -1 and with the first line {1,1,1,1}.

Answer 1

The task in the last paragraph of your question can simply be achieved by the following:

Select[UnequalTo[0]@*Det]@Map[Prepend[{1, 1, 1, 1}]]@Tuples[{1, -1}, {3, 4}]

We first generate all possible $3\times 4$ matrices with 1/-1 as entries using Tuples, then we add the row of 1s using Prepend, and finally we select those that are invertible using Select

For your general question, I think Select is what you're after.

Answer 2

Probably less efficient than Mathe172's answer, but with simpler logic:

Select[Tuples[{1, -1}, {4, 4}], Det[#] =!= 0 && #[[1]] == {1, 1, 1, 1} &]

This generates all 4x4 matrices with +/-1 entries, then selects all those with Det not zero and with the specified first row.