  In this paper, we study the connections between three concepts - the reverse
H\"older inequality for matrix-valued martingales, the well-posedness of linear
BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE
systems. In particular, we show that a linear BSDE with bmo (bounded mean
oscillation) coefficients is well-posed if and only if the stochastic
exponential of a related matrix-valued martingale satisfies a reverse H\"older
inequality. Furthermore, we give structural conditions under which these two
equivalent conditions are satisfied. Finally, we apply our results on linear
equations to obtain global well-posedness results for two new classes of
non-Markovian quadratic BSDE systems with special structure.
