This paper introduces a constrained $\ell_1$ minimization method for estimating a sparse inverse covariance matrix based on a sample of $n$ independent and identically distributed (iid) $p$-variate random variables. The estimator is shown to have several desirable properties, including fast convergence rates under the spectral norm, elementwise $\ell_\infty$ norm, and Frobenius norm. The method is implemented using linear programming and is computationally efficient, making it suitable for high-dimensional data. Numerical performance is evaluated through simulations and real data analysis, demonstrating favorable results compared to existing methods. The paper also discusses graphical model selection and provides theoretical guarantees for the estimator's performance under different moment conditions on the data.This paper introduces a constrained $\ell_1$ minimization method for estimating a sparse inverse covariance matrix based on a sample of $n$ independent and identically distributed (iid) $p$-variate random variables. The estimator is shown to have several desirable properties, including fast convergence rates under the spectral norm, elementwise $\ell_\infty$ norm, and Frobenius norm. The method is implemented using linear programming and is computationally efficient, making it suitable for high-dimensional data. Numerical performance is evaluated through simulations and real data analysis, demonstrating favorable results compared to existing methods. The paper also discusses graphical model selection and provides theoretical guarantees for the estimator's performance under different moment conditions on the data.