Obstructions to embeddings of bundles of matrix algebras in a trivial bundle

We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle A _k → X whose fiber is a complex matrix algebra M _k (?) in a trivial bundle with fiber M _kl (?) under the assumption that (k, l) = 1. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group PGL _k (?) of the bundle A _k to SL _k (?), which coincides with the first Chern class c ₁(ξ _k ) reduced modulo k under the assumption that A _k ≌ End(ξ _k ) for some vector ? _k -bundle ξ _k → X. If the first obstruction vanishes, then A _k ≌ End( $\tilde \xi _k $ ) for some vector ? ^k bundle ξ _k → X with structure group SL _k (?), and the second obstruction is c ₂( $\tilde \xi _k $ )modk ∈ H ⁴(X, ?/k?). Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.