  A modified primal-dual weak Galerkin (M-PDWG) finite element method is
designed for the second order elliptic equation in non-divergence form.
Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system
of equations resulting from the M-PDWG scheme could be equivalently simplified
into one equation involving only the primal variable by eliminating the dual
variable (Lagrange multiplier). The resulting simplified system thus has
significantly fewer degrees of freedom than the one resulting from existing
PDWG scheme. In addition, the condition number of the simplified system could
be greatly reduced when a newly introduced bilinear term in the M-PDWG scheme
is appropriately chosen. Optimal order error estimates are derived for the
numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm
respectively. Extensive numerical results are demonstrated for both the smooth
and non-smooth coefficients on convex and non-convex domains to verify the
accuracy of the theory developed in this paper.
