The additive envelope of an Ab-enriched category $A$ is defined by taking the objects as formal direct sums of objects in $A$, and morphisms as matrices of coefficients, giving an additive category. This is a universal construction.
By further taking the Karoubi envelope (i.e. formally adding images of idempotent elements), one constructs a Karoubian category called the pseudo-abelian envelope of $A$. In general, the pseudo-abelian envelope of $A$ is not abelian.