The linear regression model with a random variable (RV) measurement matrix, where the mean of the random measurement matrix has full column rank, has been extensively studied. In particular, the quasiconvexity of the maximum likelihood estimation (MLE) problem was established, and the corresponding Cramer-Rao bound (CRB) was derived, leading to the development of an efficient bisection-based algorithm known as RV-ML. In contrast, this work extends the analysis to both overdetermined and underdetermined cases, allowing the mean of the random measurement matrix to be rank-deficient. A remarkable contribution is the proof that the equivalent MLE problem is convex and satisfies strong duality, strengthening previous quasiconvexity results. Moreover, it is shown that in underdetermined scenarios, the randomness in the measurement matrix can be beneficial for estimation under certain conditions. In addition, a fast and unified implementation of the MLE solution, referred to as generalized RV-ML (GRV-ML), is proposed, which handles a more general case including both underdetermined and overdetermined systems. Extensive numerical simulations are provided to validate the theoretical findings.