using Symbolics, LinearAlgebra

@variables a[1:3, 1:3]
A = collect(a)

expand(det(A))

@variables a b c d 🍎 🍌 🍪 🥟 α β γ δ ♣ ♡ ♠ ♢

A = [a  b
     🍎 🍌]

det(A)

A = [a  b  c
     🍎 🍌 🍪
     α  β  γ]

expand(det(A))

A = [ a  b  c  d
      🍎 🍌 🍪  🥟
      α  β  γ  δ
      ♣  ♡  ♠  ♢]

expand(det(A))

@variables a[1:5,1:5]
A = collect(a)

expand(det(A))

@variables a # make this back to an ordinary scalar variable

A = [1 3
     2 4]
det(A)

A = [1 2
     2 4]
det(A)

A = randn(5,5)
det(A)

L,U = lu(A, NoPivot()) # LU without row swaps
U

prod(diag(U)) # the product of the diagonal elements of U

I₅ = I(5) * 1

det(I₅)

det([a  b
     🍎 🍌])

det([🍎 🍌
     a  b])

I₅_swapped = I₅[ [2,1,3,4,5], : ]

det(I₅_swapped)

A = rand(-3:3, 5,5)

B = A[ [2,1,3,4,5], : ]

det(A), det(B)

C = copy(A)
C[2,:] = 2*A[2,:]
C

det(A), det(C)

det(2A) / det(A)

A

[1,0,0,0,0] * [1 2 3 4 5] # = column * row = outer product

det(A + [1,0,0,0,0] * [1 2 3 4 5])

A′ = copy(A)
A′[1,:] = [1,2,3,4,5] # replace first row
A′

det(A) + det(A′)

det([ 1 2 3 
      4 5 6
      1 2 3 ])

L, U = lu(A, NoPivot()) # elimination without row swaps
U

det(A), det(U)

det([1 2 3
     4 5 6
     0 0 0])

U

diag(U)

prod(diag(U))

det(U), det(A)

@which det(A)

@which det(UpperTriangular(U))

@which det(lufact(A))

B = rand(-3:3, 5,5)

det(A), det(B)

det(A*B), det(A)*det(B)

det(inv(A)), 1/det(A)

det(A), det(A')