LinearAlgebra.det and symbolic variables

General Usage Performance
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Here’s a solution using Nemo.jl, which finishes under 3 seconds for N=100. Nemo has its own matrix type, which is a little different from Julia’s own Matrix. I also use Nemo’s rational number type QQ instead of Float64. Unfortunately, Nemo does not support sparse matrices, so I’m computing with a dense matrix, which will cause memory usage problems if N is really large.

using Nemo, LinearAlgebra

N = 100

r, z = PolynomialRing(QQ, "z")

c = fill(r(0), N) # `r(0)` is the zero polynomial. `c` is a vector of polynomials
for k in 2:N
  c[k] = r((k - 1)//(2*(N - 1)))
end

A = MatrixSpace(r, N, N)() # This is using Nemo's matrix type. Intialized to zero matrix
A[2, 1] = c[2]
for k in 3:N
  A[k, 1]   = c[k] - r(1)
  A[k, k-1] = r(1)
end

b = MatrixSpace(r, 1, N)()
b[1, N] = r(1)

matrix = MatrixSpace(r, N, N)(1) - z * A + z * MatrixSpace(r, N, 1)(r(1)) * b

display(Matrix(matrix)) # convert back to Julia's native Matrix type before calling `display`, as Nemo's printing of matrices is not as pretty

@time Determinant = det(matrix) # Nemo expands polynomials by default. No need to expand further
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