using LinearAlgebra # load this package for most of Julia's linear algebra functions

A = [1 3  1
     1 1 -1
     3 11 6]

# LU factorization (Gaussian elimination) of the matrix A, 
# passing the NoPivot() to prevent row re-ordering ("pivoting")
L, U = lu(A, NoPivot()) 
U # just show U

E1 = [ 1 0 0
      -1 1 0
      -3 0 1]

E1*A

using Symbolics

@variables s[1:3, 1:3]  # declare an bunch of symbolic variables
S = collect(s)          # & collect them into a matrix

E1 * S

@variables 😀 🥸 😭 🐸 🐶 🐨 🚗 🚛 🚜    # declare "moar funner" symbolic variables

S = [ 😀 🥸 😭       # first row = faces
      🐸 🐶 🐨       # second row = animals
      🚗 🚛 🚜 ]     # third row = Cars™

E1 * S

E2 = [1 0 0
      0 1 0
      0 1 1]

E2 * S

E2*E1*A

E2*E1*S

E2*E1*A == U

E = E2*E1

E1*E2

lu(A, NoPivot())

E^-1

A = [4  -2  -7  -4  -8
     9  -6  -6  -1  -5
    -2  -9   3  -5   2
     9   7  -9   5  -8
    -1   6  -3   9   6] # a randomly chosen 5x5 matrix

b = [-7,2,4,-4,-7] # a randomly chosen right-hand side

I(5)

I(5) * b == b

I(5) * A == A

A * I(5) == A

I

I * b

I * A == A == A * I

I * [2, 3]

B = [2 3
     4 5
     6 7]

I * B * I

I(3) * B * I(2)

I(2) * B * I(3)  # should give an error: wrong-shape matrices!

I * B * I

inv(A)

round.(inv(A) * A, 3)

C = rand(4,4)
D = rand(4,4)
inv(C*D)

inv(D)*inv(C)

inv(C)*inv(D)  # this is not the inverse of C*D

A = [1 3  1
     1 1 -1
     3 11 6]
E1 = [ 1 0 0
      -1 1 0
      -3 0 1]
E2 = [1 0 0
      0 1 0
      0 1 1]

E2*E1

E1*E2

inv(E2*E1)

E1

E2

inv(E2)

inv(E1)

L, U = lu(A, Val{false})
U

inv(E1)*inv(E2)*U

L = inv(E1)*inv(E2)

inv(E2*E1)

E1

L

inv(L)

A = [4  -2  -7  -4  -8
     9  -6  -6  -1  -5
    -2  -9   3  -5   2
     9   7  -9   5  -8
    -1   6  -3   9   6] # a randomly chosen 5x5 matrix

Ainv = A \ I(5)

Ainv - inv(A)

Ainv ≈ inv(A)

Ainv * A

A * Ainv

[A\b Ainv*b] # print the two results side-by-side