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Minimal determinant of a matrix with varied entries

I would like Sage to tell me to minimal value of the determinant of a matrix when a vary some entries over a set range.

For example, say with the 4x4 matrix (sorry for weird layout, I can't get the Latex code to work properly?):

0 1 i j

1 0 1 k

0 1 0 1

0 0 1 0

I'd like to know, for ${i,j,k} \in {0, \pm1}$, what the smallest non-trivial determinant is, and for which combination(s) of $i,j,k$ this is for.

I'm still pretty new to Sage, so I'm a bit unsure of how to do this effectively, if it is at all possible to do this?

Minimal determinant of a matrix with varied entries

I would like Sage to tell me to minimal value of the determinant of a matrix when a vary some entries over a set range.

For example, say with the 4x4 matrix A (sorry for weird layout, I can't get the Latex code to work properly?):

0 1 i j

1 0 1 k

0 1 0 1

0 0 1 0

I'd like to know, for ${i,j,k} \in {0, \pm1}$, what the smallest non-trivial determinant is, and for which combination(s) of $i,j,k$ this is for.

I'm still pretty new to Sage, so I'm a bit unsure of how to do this effectively, if it is at all possible to do this?

Side note: I know I could of course write

sage A.determinant()

and this would give me the value of the determinant in terms of $i,j,k$. And in this example that would likely be much easier.

The idea is that I'd like to do this with larger matrices (that aren't as nice as this), more variables, etc. It would obviously be nicer to have have Sage simply compute the smallest determinant and give the corresponding values for the variables than me have to plug it in.