Matrix manipulation is mainly for systems of linear equations, and this is a system of nonlinear equations. You need to either change variables to form a linear system, or use a tool designed for nonlinear systems. If the system is overdetermined (has more equations than unknowns), or there is error in the measured values you plug into the equations, there won't be an exact solution, so you'll need to pick a method that finds the best possible wrong answer, according to some criterion for "best".
In multilateration, one usually either linearizes this problem, such as a one-term Taylor expansion around a point, and solves by QR decomposition or SVD; or uses a nonlinear solver like Levenberg–Marquardt, Nelder–Mead, or BFGS. These all have to be started off with an initial guess, which may be obtained by something such as the transformation from intersecting spheres to planes and lines described in this answer.
For processing the data from GPS itself, or other navigation satellite systems, there is a very good tutorial with lots of equations at the European Space Agency's Navipedia.
In this case, all your equations happen to be polynomials (once you move the $v$ terms to the other side and square), so you can do fancy tricks with modern computer algebra tools like Gröbner bases, but this is very uncommon in the navigation and geolocation literature.