n=5

A = [ 3 21 8 19 6
    5 -4 11 14 13
     √5 -2 2.5 0 0
      0 0 0 0 1
     2 99 5 π π]

m1 = A[2:5,1] / A[1,1]

E1 = eye(5)
E1[2:5,1] = -m1

A2 = E1 * A

m2 = A2[3:5,2]/A2[2,2]

E2 = eye(5)
E2[3:5,2]= -m2

E2

A3 = E2 * A2

m3 = A3[4:5,3]/A3[3,3]

E4 = eye(5)
E4[4:5,3] = - m3


A4 = E4 * A3

P = eye(5)
P[[4 5],:] = P[[5,4],:]

A4P = P * A4

E1

inv(E1)

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

# LU factorization (Gaussian elimination) of the matrix A, 
# passing the ( will go away) option Val{false} to prevent row re-ordering
L, U = lu(A, Val{false}) 
U # just show U

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

Int.(inv(E1))  ## What does this mean?

E1 * A

inv(E1) *  (E1 * A)

E1*A

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

E2*E1*A

E2*E1*A == U

E2*E1

E1*E2

E1

inv(E2)

inv(E1)

inv(E1)*inv(E2)*U == A

L = inv(E1)*inv(E2)

inv(E2*E1)

L

inv(E2*E1)

E1

E2

L

inv(L)

E1

inv(E1)

E2

inv(E2)

L

inv(L)

E1 # differs from I in column 1

E2 # differs from I in column 2

L# differs from I in more than one column L

E2 * E1