In many core systems, run-time scheduling decisions, such as task migration, core activations/deactivations, voltage/frequency scaling, etc., are typically used to optimize the resource usages. Such run-time decisions change the power consumption, which can in turn result in transient temperatures much higher than any steady-state scenarios. Therefore, to be thermally safe, it is important to evaluate the transient peaks before making resource management decisions. This paper presents a method for computing these transient peaks in just a few milliseconds, which is suited for run-time usage. This technique works for any compact thermal model consisting in a system of first-order differential equations, for example, RC thermal networks. Instead of using regular numerical methods, our algorithm is based on analytically solving the differential equations using matrix exponentials and linear algebra. This results in a mathematical expression which can easily be analyzed and differentiated to compute the maximum transient temperatures. Moreover, our method can also be used to efficiently compute all transient temperatures for any given time resolution without accuracy losses. We implement our solution as an open-source tool called MatEx. Our experimental evaluations show that the execution time of MatEx for peak temperature computation can be bounded to no more than 2.5 ms for systems with 76 thermal nodes, and to no more than 26.6 ms for systems with 268 thermal nodes, which is three orders of magnitude faster than the state-of-the-art for the same settings.