Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing analytical methods for solving FFSME are based on Vec-operator and Kronecker product. Nevertheless, these methods are only applicable for nonnegative fuzzy numbers, which limits the applications of the existing methods. Thus, this paper proposes a new numerical method for solving arbitrary Trapezoidal FFSME (TrFFSME), which includes near-zero trapezoidal fuzzy numbers to overcome this limitation. The TrFFSME is converted to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication operations. Then the non-linear system is solved using a newly developed two-stage algorithm. In the first stage algorithm, initial values are determined. Subsequently, the second stage algorithm obtains all possible finite fuzzy solutions. A numerical example is solved to illustrate the proposed method. Besides, this proposed method can solve other forms of fuzzy matrix equations and produces finite fuzzy and non-fuzzy solutions compared to the existing methods.

Artificial intelligence is increasingly present in educational settings, yet its integration remains fragmented, lacking coherent understanding of how teachers and AI can effectively collaborate. This systematic review of 169 peer-reviewed studies (2009–2024) introduces the AI-Human Synergy Framework, a model for categorizing educational AI based on autonomy and collaboration, empirically validated through systematic coding with exceptional inter-rater reliability (κ = 0.972). Our analysis reveals significant automation bias: 45.56% of implementations substitute human functions, while only 4.73% achieve genuine human-AI partnership—a 10:1 ratio. Through rigorous analysis across diverse AI methods—from Predictive Analytics to Generative AI—we demonstrate that educators are predominantly positioned as data interpreters rather than collaborative partners. While recent work (2023–2024) shows promising shifts toward Augmentation (62.5%), true Synergy remains critically underrepresented. This framework provides the first empirical quantification of the substitution-synergy gap and offers actionable implementation tools, underscoring the urgent need to prioritize collaborative design over automation as Generative AI reshapes educational practice.

In the fuzzy literature, researchers have applied the concept of Vec-operator and Kronecker product for solving arbitrary Fuzzy Matrix Equations (FME). However, this approach is limited to positive or negative FMEs and cannot be applied to FMEs with near-zero fuzzy numbers. Therefore, this paper proposes a new analytical method for solving a family of arbitrary FMEs. The proposed method is able to solve Arbitrary Generalized Trapezoidal Fully Fuzzy Sylvester Matrix Equations (AGTrFFSME), in addition to many unrestricted FMEs such as Sylvester, Lyapunov and Stein fully fuzzy matrix equations with arbitrary triangular or trapezoidal fuzzy numbers. The proposed method thus fruitfully removes the sign restriction imposed by researchers and is, therefore, better to use in several engineering and scientific applications. The AGTrFFSME is converted to a system of non-linear equations, which is reduced using new multiplication operations between trapezoidal fuzzy numbers. The feasibility conditions are introduced to distinguish between fuzzy and non-fuzzy solutions to the AGTrFFSME.

Linear systems often involve coefficients that are uncertain or imprecise due to inherent variability and vagueness in the data. In scenarios where only approximate or vague knowledge of the system parameters is available, traditional fuzzy logic is commonly employed. However, conventional fuzzy logic may be inadequate when defining a membership degree with a single, precise value proves difficult. In such cases, Single-Valued Trapezoidal Neutrosophic Numbers (SVTrNNs) offer a more suitable framework, as they account for indeterminacy, alongside truth and falsity. The solution of Single-Valued Trapezoidal Neutrosophic Linear Equations (SVTrNLEs) was explored in this study using an embedding approach. The approach reformulates the SVTrNLEs into an equivalent crisp linear system, enabling the application of conventional solution methods. The solution was then obtained using either the matrix inversion method or the gradient descent optimization algorithm implemented in PyTorch. The robustness and adaptability of gradient-based optimization techniques were thoroughly assessed. The learning process minimizes the residual error iteratively, with convergence behaviour and numerical stability analyzed across various parameter configurations. The results demonstrate rapid convergence, proximity to exact solutions, and significant robustness to parameter variability, highlighting the efficacy of gradient descent for solving uncertain linear systems. These findings provide a foundation for the extension of gradient-based methods to more complex systems and broader applications. Furthermore, the existence and uniqueness of the neutrosophic solution to an n × n linear system were rigorously analyzed, with numerical examples provided to assess the reliability and efficiency of the proposed methods.