Summary: In this paper, we consider the problem of estimating simultaneously low-rank and row-wise sparse matrices from nested linear measurements, where the linear operator consists of the product of a linear operator \(\mathcal W\) and a matrix \(\boldsymbol{\Psi}\). Leveraging the nested structure of the measurement operator, we propose a computationally efficient two-stage algorithm for estimating the simultaneously structured target matrix. Assuming that \(\mathcal{W}\) is a restricted isometry for low-rank matrices and \(\boldsymbol{\Psi}\) is a restricted isometry for row-wise sparse matrices, we establish an accuracy guarantee that holds uniformly for all sufficiently low-rank and row-wise sparse matrices with high probability. Furthermore, using standard tools from information theory, we establish a minimax lower bound for estimation of simultaneously low-rank and row-wise sparse matrices from linear measurements that need not be nested. The accuracy bounds established for the algorithm, that also serve as a minimax upper bound, differ from the derived minimax lower bound merely by a polylogarithmic factor of the dimensions. Therefore, the proposed algorithm is nearly minimax optimal. We also discuss some applications of the proposed observation model and evaluate our algorithm through numerical simulation.