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Cloud computing paradigm provides an alternative and economical service for resource-constrained clients to perform large-scale data computation. Since large matrix determinant computation (DC) is ubiquitous in the fields of science and engineering, a first step is taken in this paper to design a protocol that enables clients to securely, verifiably, and efficiently outsource DC to a malicious cloud. The main idea to protect the privacy is employing some transformations on the original matrix to get an encrypted matrix which is sent to the cloud; and then transforming the result returned from the cloud to get the correct determinant of the original matrix. Afterwards, a randomized Monte Carlo verification algorithm with one-sided error is introduced, whose superiority in designing inexpensive result verification algorithm for secure outsourcing is well demonstrated. In addition, it is analytically shown that the proposed protocol simultaneously fulfills the goals of correctness, security, robust cheating resistance, and high-efficiency. Extensive theoretical analysis and experimental evaluation also show its high-efficiency and immediate practicability. It is hoped that the proposed protocol can shed light in designing other novel secure outsourcing protocols, and inspire powerful companies and working groups to finish the programming of the demanded all-inclusive scientific computations outsourcing software system. It is believed that such software system can be profitable by means of providing large-scale scientific computation services for so many potential clients.