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In this paper, we introduce novel fast matrix inversion algorithms thatleverage triangular decomposition and recurrent formalism, incorporatingStrassen's fast matrix multiplication. Our research places particular emphasison triangular matrices, where we propose a novel computational approach basedon combinatorial techniques for finding the inverse of a general non-singulartriangular matrix. Unlike iterative methods, our combinatorial approach for(block) triangular-type matrices enables direct computation of the matrixinverse through a nonlinear combination of carefully selected combinatorialentries from the initial matrix. This unique characteristic makes our proposedmethod fully parallelizable, offering significant potential for efficientimplementation on parallel computing architectures. Our approach demonstratesintriguing features that allow the derivation of recurrent relations forconstructing the matrix inverse. By combining the (block) combinatorialapproach, with a recursive triangular split method for inverting triangularmatrices, we develop potentially competitive algorithms that strike a balancebetween efficiency and accuracy. We provide rigorous mathematical proofs of thenewly presented method. Additionally, we conduct extensive numerical tests toshowcase its applicability and efficiency. The comprehensive evaluation andexperimental results presented in this paper confirm the practical utility ofour proposed algorithms, demonstrating their superiority over classicalapproaches in terms of computational efficiency.

Combinatorial and Recurrent Approaches for Efficient Matrix Inversion: Sub-cubic algorithms leveraging Fast Matrix products