  Current experimental data on neutrino oscillations are consistent with the
tri-bimaximal mixing. If future experimental data will determine a non-zero
$V_{e3}$ and/or find CP violations in neutrino oscillations, there is the need
to modify the mixing pattern. We find that a simple neutrino mass matrix,
resulting from $A_4$ family symmetry breaking with residual $Z_3$ and $Z_2$
discrete symmetries respectively for the Higgs sectors generating the charged
lepton and neutrino mass matrices, can satisfy the required modifications. The
neutrino mass matrix is minimally modified with just one additional complex
parameter compared with the one producing the tri-bimaximal mixing. In this
case, the CP violating Jarlskog factor $J$ has a simple form
($|J|=|V_{e1}V_{e3}|/2\sqrt{3}$ for real neutrino mass matrix), and also
$V_{\mu i} = 1/\sqrt{3}$. We also discuss how this mixing matrix can be tested
experimentally.
