 Solution

If $A=\mathrm{Im}V={\mathbb{R}}^n$, the symmetric covariance matrix $V$ is positive definite. For every nonzero $H$, the scalar $H\cdot P_1$ is normal with variance $H^{T}VH>0$, so it has positive probability of being negative. It cannot be an arbitrage payoff. The zero portfolio provides no strict gain, proving absence of arbitrage.