In this study, the authors analyse the instability of continuous-time Markov jump linear systems. Although there exist several effective criteria for the stability of Markov jump linear systems, there is a lack of methodologies for verifying their instability. In this study, they present a novel criterion for the exponential mean instability of Markov jump linear systems. The main tool of the authors' analysis is an auxiliary Markov jump linear system, which results from taking the Kronecker products of the given system matrices and a set of appropriate matrix weights. They show that the problem of finding matrix weights for tighter instability analysis can be transformed into the spectral optimisation problem on an affine matrix family, which can be efficiently solved by gradient-based non-smooth optimisation algorithms. They confirm the effectiveness of the proposed methods by numerical examples.