Note for posterity: the go-to module for this stuff is PDL::LinearAlgebra. The obvious solver for Ax=B would be "msolve", and the notes for that give which LAPACK function is used "under the hood". For solving many times, you'd want to do a decomposition (quite possibly an LU) first, then reuse it. Take a look!

LAPACK is a huge and powerful library for linear-algebra stuff, and getting to know it will be useful, regardless of which language environment you use - I don't think there is a single language environment that doesn't have a binding for it.