Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Solving over a polynomial ring:
sage: x = polygen(QQ, ’x’)
sage: A = matrix(2, [x,2*x,-5*x^2+1,3])
sage: v = vector([3,4*x - 2])
sage: X = A \ v
sage: X
((-8*x^2 + 4*x + 9)/(10*x^3 + x), (19*x^2 - 2*x - 3)/(10*x^3 + x))
sage: A * X == v
True
Solving some systems over Z/nZ:
sage: A = Matrix(Zmod(6), 3, 2, [1,2,3,4,5,6])
sage: B = vector(Zmod(6), [1,1,1])
sage: A.solve_right(B)
(5, 1)
sage: B = vector(Zmod(6), [5,1,1])
sage: A.solve_right(B)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions
sage: A = Matrix(Zmod(128), 2, 3, [23,11,22,4,1,0])
sage: B = Matrix(Zmod(128), 2, 1, [1,0])
sage: A.solve_right(B)
[ 1]
[124]
[ 1]
Solving a system over the p-adics:
sage: k = Qp(5,4)
sage: a = matrix(k, 3, [1,7,3,2,5,4,1,1,2]); a
[ 1 + O(5^4) 2 + 5 + O(5^4) 3 + O(5^4)]
[ 2 + O(5^4) 5 + O(5^5) 4 + O(5^4)]
[ 1 + O(5^4) 1 + O(5^4) 2 + O(5^4)]
sage: v = vector(k, 3, [1,2,3])
sage: x = a \ v; x
(4 + 5 + 5^2 + 3*5^3 + O(5^4), 2 + 5 + 3*5^2 + 5^3 + O(5^4), 1 + 5 + O(5^4))
sage: a * x == v
True
Solving a system of linear equation symbolically using symbolic matrices:
sage: var(’a,b,c,d,x,y’)
(a, b, c, d, x, y)
sage: A=matrix(SR,2,[a,b,c,d]); A
[a b]
[c d]
sage: result=vector(SR,[3,5]); result
(3, 5)
sage: soln=A.solve_right(result)
sage: soln
(-b*(3*c/a - 5)/(a*(b*c/a - d)) + 3/a, (3*c/a - 5)/(b*c/a - d))
sage: (a*x+b*y).subs(x=soln[0],y=soln[1]).simplify_full()
3
sage: (c*x+d*y).subs(x=soln[0],y=soln[1]).simplify_full()
5
sage: (A*soln).apply_map(lambda x: x.simplify_full())
(3, 5)
256 Chapter 7. Base class for matrices, part 2