┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.5.beta1, Release Date: 2024-08-03              │
│ Using Python 3.12.4. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
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┃ Warning: this is a prerelease version, and it may be unstable.     ┃
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sage: n = 8
sage: A = matrix.random(QQ, n, n)
sage: A
[   0    1    0  1/2   -2   -1    0   -1]
[   0 -1/2   -2    2    1    0    0    2]
[   0    1  1/2    1    0    0    0    0]
[ 1/2    2    0   -1    2    0    1    2]
[  -1   -1    1   -1    1 -1/2   -2    1]
[   2   -1    1    0    0 -1/2    0   -1]
[   2   -1    0    0    1    0    0   -2]
[   2    0    0   -2   -2    1    1    2]

sage: A.inverse()
[ 1764/11299  1336/11299  3112/11299   574/11299  1448/11299  -332/11299  4240/11299  2322/11299]
[ 3032/11299  -522/11299  5144/11299  1704/11299  1464/11299 -5080/11299  4162/11299  1224/11299]
[-3520/11299 -2256/11299  3270/11299  -428/11299  -686/11299  5838/11299 -5130/11299  -944/11299]
[-1272/11299  1650/11299  4520/11299 -1490/11299 -1121/11299  2161/11299 -1597/11299  -752/11299]
[-1738/11299    16/11299  -504/11299  1766/11299    85/11299   199/11299   998/11299 -1596/11299]
[-4806/11299 -1334/11299  8124/11299 -3178/11299   -25/11299 -6705/11299  4359/11299  3128/11299]
[-3500/11299 -1216/11299 -6892/11299  1372/11299 -6460/11299  7474/11299 -8054/11299 -2993/11299]
[ -621/11299  1605/11299   288/11299   605/11299  1517/22598  4615/22598 -5983/22598   912/11299]

sage: A.det()
11299/16

sage: v = Matrix.random(QQ, n, 1)
sage: v
[-1]
[ 0]
[ 1]
[-1]
[ 1]
[ 0]
[-1]
[-1]

sage: A.solve_right(v)
[-4340/11299]
[-3514/11299]
[12606/11299]
[ 8510/11299]
[  151/11299]
[ 8596/11299]
[ -177/11299]
[ 3142/11299]

sage: A.rref()
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]

sage: A = matrix.random(QQ, n, n)
sage: A.rref()
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]

sage: A = matrix.random(QQ, n, n)
sage: A.rref()
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]

sage: A = matrix.random(GF(2), n, n)
sage: A.rref()
[1 0 0 0 1 0 0 1]
[0 1 0 0 1 0 0 0]
[0 0 1 0 0 0 0 1]
[0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]

sage: reset()

sage: field = GF(next_prime(2**21))
sage: field
Finite Field of size 2097169

sage: mat1 = matrix.random(field, 2000, 2000)
sage: mat2 = matrix.random(field, 2000, 2000)
sage: %time mat = mat1 * mat2
CPU times: user 2.24 s, sys: 475 ms, total: 2.71 s
Wall time: 2.69 s

sage: mat1 = matrix.random(field, 4000, 4000)
sage: mat2 = matrix.random(field, 4000, 4000)
sage: %time mat = mat1 * mat2
CPU times: user 14.6 s, sys: 1.61 s, total: 16.2 s
Wall time: 16 s