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We study resonances for Jacobi operators on the half lattice with matrixvalued coefficient and finitely supported perturbations. We describe aforbidden domain, the geometry of resonances and their asymptotics when themain coefficient of the perturbation (determining its length of support) goesto zero. Moreover, we show that\\ 1) Any sequence of points on the complexplane can be resonances for some Jacobi operators. In particular, themultiplicity of a resonance can be any number.\\ 2) The Jost determinantcoincides with the Fredholm determinant up to the constant.\\ 3) The S-matrixon the a.c spectrum determines the perturbation uniquely.\\ 5) The value of theJost matrix at any finite sequence of points on the a.c spectrum determine theJacobi matrix uniquely. The length of this sequence is equals to the upperpoint of the support perturbation.

Resonances for vector-valued Jacobi operators on half-lattice