  In this article we further the study of non-commutative motives. Our main
result is the construction of a simple model, given in terms of infinite
matrices, for the suspension in the triangulated category of non-commutative
motives. As a consequence, this simple model holds in all the classical
invariants such as Hochschild homology, cyclic homology and its variants
(periodic, negative, ...), algebraic K-theory, topological Hochschild homology,
topological cyclic homology, ...
