with(linalg):
A :=  matrix(5,5,[1,1,1,0,0, 
                  1,2,0,1,0,
                  3,2,0,0,1,
                  3,0,2,0,3,
                  0,2,0,0,1]):
b := vector(5,[ 2,1,0,1,0 ]):
A1:=evalm(A): 
A1[1,5]:=b[1]:A1[2,5]:=b[2]:A1[3,5]:=b[3]:A1[4,5]:=b[4]:A1[5,5]:=b[5]:
evalm(A1),evalm(inverse(A1)),inverse(A),det(A),det(A1);

                         [ 0      0    1/3      0      -1/3]
[1    1    1    0    2]  [                                 ]
[                     ]  [ 0      0     0       0      1/2 ]
[1    2    0    1    1]  [                                 ]
[                     ]  [                              13 ]
[3    2    0    0    0], [-1/3    0    -5/9    2/3      -- ],
[                     ]  [                              18 ]
[3    0    2    0    1]  [                                 ]
[                     ]  [-2/3    1    -4/9    1/3     -2/9]
[0    2    0    0    0]  [                                 ]
                         [2/3     0    1/9     -1/3    -4/9]

    [ 0      0    1/3      0      -1/3]
    [                                 ]
    [1/4     0    1/24    -1/8    1/3 ]
    [                                 ]
    [3/4     0    -3/8    1/8      0  ]
    [                                 ]
    [              -5                 ], -24, 18
    [-1/2    1     --     1/4     -1/3]
    [              12                 ]


with(linalg):
A :=  matrix(5,5,[ 2,-1, 1, 1, 0, 
                  -1, 2, 0, 2, 0,
                   2, 4,-1, 1, 1,
                   2, 0, 1, 0, 3,
                   0, 3, 0, 1, 3]):
b := vector(5,[ 2,1,0,1,0 ]):
A1:=evalm(A): 
A1[1,5]:=b[1]:A1[2,5]:=b[2]:A1[3,5]:=b[3]:A1[4,5]:=b[4]:A1[5,5]:=b[5]:
evalm(A1),evalm(inverse(A1)),inverse(A),det(A),det(A1);
[ 2    -1     1    1    2]  [ 1     -1       0       -1        1  ]
[                        ]  [                                     ]
[-1     2     0    2    1]  [-1     4/5     1/5      6/5     -4/5 ]
[                        ]  [                                     ]
[ 2     4    -1    1    0], [ 1    -6/5     -4/5    -4/5     11/5 ],
[                        ]  [                                     ]
[ 2     0     1    0    1]  [ 3    -12/5    -3/5    -18/5    17/5 ]
[                        ]  [                                     ]
[ 0     3     0    1    0]  [-3    16/5     4/5     19/5     -21/5]

    [  1       -1     0     -1        1  ]
    [                                    ]
    [-23/2     12     3    29/2     -31/2]
    [                                    ]
    [-49/2     26     6    63/2     -67/2], 2, -5
    [                                    ]
    [ 12      -12    -3     -15      16  ]
    [                                    ]
    [15/2      -8    -2    -19/2    21/2 ]


with(linalg):
A :=  matrix(5,5,[ 1,-1, 1, 1, 0, 
                  -1, 2, 0, 2, 0,
                   1, 1,-1, 1, 1,
                   2, 0, 1, 0, 3,
                   0, 1, 0, 1, 1]):
b := vector(5,[ 0,-1,0,0,-1 ]):
A1:=evalm(A): 
A1[1,5]:=b[1]:A1[2,5]:=b[2]:A1[3,5]:=b[3]:A1[4,5]:=b[4]:A1[5,5]:=b[5]:
evalm(A1),evalm(inverse(A1)),inverse(A),det(A),det(A1);

[ 1    -1     1    1     0]  [ 0      -1/4    1/4     1/4     1/4 ]
[                         ]  [                                    ]
[-1     2     0    2    -1]  [-1/2    1/2      0      1/2     -1/2]
[                         ]  [                                    ]
[ 1     1    -1    1     0], [ 0      1/2     -1/2    1/2     -1/2],
[                         ]  [                                    ]
[ 2     0     1    0     0]  [1/2     1/4     1/4     -1/4    -1/4]
[                         ]  [                                    ]
[ 0     1     0    1    -1]  [ 0      3/4     1/4     1/4     -7/4]

    [ 0       1      2/3     2/3     -8/3 ]
    [                                     ]
    [-1/2    3/2     1/3     5/6     -17/6]
    [                                     ]
    [ 0       1      -1/3    2/3     -5/3 ], 6, -8
    [                                     ]
    [1/2     -1/2     0      -1/2     3/2 ]
    [                                     ]
    [ 0       -1     -1/3    -1/3     7/3 ]