A =
     1     4    -1
     3    11    -8
    -2    -4    23
I=eye(3)
I =
     1     0     0
     0     1     0
     0     0     1
B=[A I]
B =
     1     4    -1     1     0     0
     3    11    -8     0     1     0
    -2    -4    23     0     0     1
E=I;
E(2,1)=-3; E(3,1)=2
E =
     1     0     0
    -3     1     0
     2     0     1
B=E*B
B =
     1     4    -1     1     0     0
     0    -1    -5    -3     1     0
     0     4    21     2     0     1
E=I; E(3,2)=4
E =
     1     0     0
     0     1     0
     0     4     1
B=E*B
B =
     1     4    -1     1     0     0
     0    -1    -5    -3     1     0
     0     0     1   -10     4     1

E=I; E(2,3)=5; E(1,3)=1
E =
     1     0     1
     0     1     5
     0     0     1
C=E*B
C =
     1     4     0    -9     4     1
     0    -1     0   -53    21     5
     0     0     1   -10     4     1
C(2,:)=-C(2,:)
C =
     1     4     0    -9     4     1
     0     1     0    53   -21    -5
     0     0     1   -10     4     1
E=I; E(1,2)=-4
E =
     1    -4     0
     0     1     0
     0     0     1
C=E*C
C =
     1     0     0  -221    88    21
     0     1     0    53   -21    -5
     0     0     1   -10     4     1
Ai=C(:,4:6)
Ai =
  -221    88    21
    53   -21    -5
   -10     4     1
A*Ai
ans =
     1     0     0
     0     1     0
     0     0     1
[A I]
ans =
     1     4    -1     1     0     0
     3    11    -8     0     1     0
    -2    -4    23     0     0     1
B
B =
     1     4    -1     1     0     0
     0    -1    -5    -3     1     0
     0     0     1   -10     4     1
C
C =
     1     0     0  -221    88    21
     0     1     0    53   -21    -5
     0     0     1   -10     4     1
[L,U]=lunp(A)
L =
     1     0     0
     3     1     0
    -2    -4     1
U =
     1     4    -1
     0    -1    -5
     0     0     1
inv(L)
ans =
    1.0000    0.0000    0.0000
   -3.0000    1.0000         0
  -10.0000    4.0000    1.0000