In the asymptotic stability (AS) analysis of interval matrix systems, some results are available that operate only as sufficient conditions, based on a unique test matrix, adequately built from the interval matrix. Our note reveals the complete role of this test matrix for fully characterizing the componentwise asymptotic stability (CWAS) of interval matrix systems. CWAS is a special type of AS which ensures the flow invariance of certain time-dependent sets with respect to the state-space trajectories. Hence, the sufficient conditions for AS get a new and deeper meaning by their reformulation as necessary and sufficient conditions with respect to the stronger property of CWAS.