  For a given extension $A \subset E$ of associative algebras we describe and
classify up to an isomorphism all $A$-complements of $E$, i.e. all subalgebras
$X$ of $E$ such that $E = A + X$ and $A \cap X = \{0\}$. Let $X$ be a given
complement and $(A, \, X, \, \triangleright, \triangleleft, \leftharpoonup,
\rightharpoonup \bigl)$ the canonical matched pair associated with the
factorization $E = A + X$. We introduce a new type of deformation of the
algebra $X$ by means of the given matched pair and prove that all
$A$-complements of $E$ are isomorphic to such a deformation of $X$. Several
explicit examples involving the matrix algebra are provided.
