Others have pointed out that matrix inversion is O(n^3) and hence computationally infeasible. It is also worth considering that the conditioning of X^t X, k(X^t X) can be as large as k(X)^2, so solving in this way can be very unstable.
While this is correct, you do not need to compute the inverse to solve this kind of least squares problem (and you do not need the inverse to solve any linear systems of equations).
