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Starting from an already defined matrix, use its change_ring method to produce a matrix over a different ring.

Suppose we have defined our initial matrix:

sage: A = matrix([[25*x^3 + 50*x^2 + 1], [18*x + 11]])
sage: A
[25*x^3 + 50*x^2 + 1]
[          18*x + 11]

Consider it over the polynomial ring over integers modulo two:

sage: B = A.change_ring(GF(2)['x'])
sage: B
[x^3 + 1]
[      1]

If needed, move entries from there to polynomials over the integers:

sage: C = B.change_ring(ZZ['x'])
sage: C
[x^3 + 1]
[      1]

One of them behaves modulo two, the other does not:

sage: 3 * B
[x^3 + 1]
[      1]

sage: 3 * C
[3*x^3 + 3]
[        3]