Determinant of an integer matrix is float

General Usage
3

Nowadays, Julia includes an integer-determinant algorithm (the Bareiss algorithm), but only uses it by default for BigInt matrices (since for fixed-width integer types like the default Int a determinant will quickly overflow unless the matrix is tiny).

julia> using LinearAlgebra

julia> M = BigInt[10 20; 30 40]
2×2 Matrix{BigInt}:
 10  20
 30  40

julia> det(M) # exact BigInt determinant
-200

You can invoke the Bareiss algorithm explicitly for other integer types by calling LinearAlgebra.det_bareiss, but beware that it can easily overflow for larger matrices:

julia> LinearAlgebra.det_bareiss([10 20; 30 40]) # exact Int determinant
-200

For example, with a 100 \times 100 matrix of random ±1 entries:

julia> A = rand([-1,1], 100,100);

julia> det(A) # floating-point approximation: fast and reasonably accurate
7.014043363270618e77

julia> det(BigInt.(A)) # exact
701404336326996708202480358186962679200983525458711992094310103304386652405760

julia> LinearAlgebra.det_bareiss(A) # overflows — bogus answer
746

PS. I just pushed a PR to clarify the det documentation: document exact BigInt determinants by stevengj · Pull Request #53579 · JuliaLang/julia · GitHub

12 Likes