How do I save a rational matrix for Mathematica and Matlab to read?

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You want to use the method _mathematica_init_(). For matlab, I haven't seen _matlab_init_() method but there is a _octave_init_() method and according to Octave website "The Octave syntax is largely compatible with Matlab" :

sage: A = sin(x)
sage: A._mathematica_init_()
'Sin[x]'
sage: A._octave_init_()
'sin(x)'

sage: B = matrix(3, range(9))
sage: B._mathematica_init_()
'{{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}'
sage: B._octave_init_()
'[0 1 2]\n[3 4 5]\n[6 7 8]'

You have a bunch of those methods:

sage: [method for method in dir(A) if method.endswith('_init_')]
['_axiom_init_',
 '_fricas_init_',
 '_gap_init_',
 '_giac_init_',
 '_gp_init_',
 '_interface_init_',
 '_kash_init_',
 '_macaulay2_init_',
 '_magma_init_',
 '_maple_init_',
 '_mathematica_init_',
 '_maxima_init_',
 '_maxima_lib_init_',
 '_octave_init_',
 '_pari_init_',
 '_polymake_init_',
 '_r_init_',
 '_singular_init_']

If you have mathematica installed, you may use it directly from sage by doing things like below. For me it does not work, because I do not have mathematica installed:

sage: Bm = mathematica(B)
...
TypeError: unable to start mathematica: End Of File (EOF). Exception style platform.
sage: Bm.Inverse() # or something like this

Conversely, with sage 8.6 because of ticket #25501 you may do:

┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 8.6, Release Date: 2019-01-15                     │
│ Using Python 2.7.15. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
sage: from sage.symbolic.integration.external import symbolic_expression_from_mathematica_string
sage: symbolic_expression_from_mathematica_string('Sin[x]')
sin(x)
sage: symbolic_expression_from_mathematica_string('{{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}')
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]