Let's do some Gaussian Elimination!

A = matrix(QQbar, 5,5, [1..25])
A

[ 1  2  3  4  5]
[ 6  7  8  9 10]
[11 12 13 14 15]
[16 17 18 19 20]
[21 22 23 24 25]

b = vector([2,3,5,7,11])
b

(2, 3, 5, 7, 11)

A1 = A.augment(b, subdivide=True)

A1

[ 1  2  3  4  5| 2]
[ 6  7  8  9 10| 3]
[11 12 13 14 15| 5]
[16 17 18 19 20| 7]
[21 22 23 24 25|11]

A2 = A1.with_added_multiple_of_row(1,0,-6); A2

[  1   2   3   4   5|  2]
[  0  -5 -10 -15 -20| -9]
[ 11  12  13  14  15|  5]
[ 16  17  18  19  20|  7]
[ 21  22  23  24  25| 11]

A3 = A2.with_added_multiple_of_row(2,0,-11); A3

[  1   2   3   4   5|  2]
[  0  -5 -10 -15 -20| -9]
[  0 -10 -20 -30 -40|-17]
[ 16  17  18  19  20|  7]
[ 21  22  23  24  25| 11]

A4 = A3.with_added_multiple_of_row(3,0,-16); A4

[  1   2   3   4   5|  2]
[  0  -5 -10 -15 -20| -9]
[  0 -10 -20 -30 -40|-17]
[  0 -15 -30 -45 -60|-25]
[ 21  22  23  24  25| 11]

A5 = A4.with_added_multiple_of_row(4,0,-21); A5

[  1   2   3   4   5|  2]
[  0  -5 -10 -15 -20| -9]
[  0 -10 -20 -30 -40|-17]
[  0 -15 -30 -45 -60|-25]
[  0 -20 -40 -60 -80|-31]

A1.rref()

[ 1  0 -1 -2 -3| 0]
[ 0  1  2  3  4| 0]
[ 0  0  0  0  0| 1]
[ 0  0  0  0  0| 0]
[ 0  0  0  0  0| 0]

z = vector([0,0,0,0,0])
A.augment(z, subdivide=True).rref()

[ 1  0 -1 -2 -3| 0]
[ 0  1  2  3  4| 0]
[ 0  0  0  0  0| 0]
[ 0  0  0  0  0| 0]
[ 0  0  0  0  0| 0]