METAL COMPOSITE HYDROXIDE AND METHOD FOR PRODUCING SAME, POSITIVE ELECTRODE ACTIVE MATERIAL FOR NON-AQUEOUS ELECTROLYTE SECONDARY BATTERY AND METHOD FOR PRODUCING SAME, AND NON-AQUEOUS ELECTROLYTE SECONDARY BATTERY

A metal composite hydroxide represented by a general formula (1): NiCoMnM(OH) (where 0.02≤x≤0.3, 0.02≤y≤0.3, 0≤z≤0.05, and −0.5≤α≤0.5 are satisfied and M is at least one element selected from the group consisting of Mg, Ca, Al, Si, Fe, Cr, V, Mo, W, Nb, Ti, and Zr), in which the metal composite hydroxide contains a first particle having a core portion inside the particle and a shell portion formed around the core portion and [(D90−D10)/MV] is 0.80 or more.