Good free calculator for manipulating symbolic matrices of 6x6 and larger?

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I have a symbolic 6-by-6 matrix containing the (pseudo-Riemannian) metric coefficients for a 6-dimensional manifold. I am trying to calculate the inverse matrix in service of working out the Christoffel symbols. It seems the size of the matrix is too big for Symbolab and Wolfram Alpha to work out, and I'm not sure if I legally can find a free copy of Mathematica or Matlab to download to my laptop (my university doesn't offer it free). Any programs you know of that could handle the computational load of a symbolic matrix this size?