The goal of this study is to solve a non-linear matrix equation of the form $ \mathcal{X} = \mathcal{Q} + \sum\limits_{i = 1}^{m} \mathcal{B}_{i}^{*}\mathcal{G} (\mathcal{X})\mathcal{B}_{i} $, where $ \mathcal{Q} $ is a Hermitian positive definite matrix, $ \mathcal{B}_{i}^{*} $ stands for the conjugate transpose of an $ n\times n $ matrix $ \mathcal{B}_{i} $ and $ \mathcal{G} $ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $ \mathcal{G}(O) = O $. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $ \mathcal{FG} $-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.