Cloud computing enables resource-constrained clients to economically outsource their huge computation workloads to a cloud server with massive computational power. This promising computing paradigm inevitably brings in new security concerns and challenges, such as input/output privacy and result verifiability. Since matrix inversion computation (MIC) is a quite common scientific and engineering computational task, we are motivated to design a protocol to enable secure, robust cheating resistant, and efficient outsourcing of MIC to a malicious cloud in this paper. The main idea to protect the privacy is employing some transformations on the original matrix to get a encrypted matrix which is sent to the cloud; and then transforming the result returned from the cloud to get the correct inversion of the original matrix. Next, a randomized Monte Carlo verification algorithm with one-sided error is employed to successfully handle result verification. In this paper, the superiority of this novel technique in designing inexpensive result verification algorithm for secure outsourcing is well demonstrated. We analytically show 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.