  This paper describes a new kind of inverse for elements in associative ring,
that is the complete inverse, as the unique solution of a certain set of
equations. This inverse exists for an element $a$ if and only if the Drazin
inverse of $a$ exists. We also show that by plugging in $-k$ for $k$ in the
$\mathcal{P}$-canonical form of a square matrix $A$, we get the
$\mathcal{P}$-canonical form of the complete inverse of $A$.
