Matrix fast power for linear recurrences (4 patterns), 4 semiring templates (count/bool/min-plus/max-plus), matrix+segment tree for range-add Fibonacci queries, Kitamasa O(k²logn) and NTT O(k log²k logn).
Gauss elimination: real/mod/mod-2 templates (solve, det, inv, rank); incremental online RREF; Extended Lights Out, CF 1344F abstract GF(2) modelling, 2024 ICPC Kunming interactive tree queries.
XOR linear basis: full template with kth, range basis, normalisation; Nowcoder median example; CF 1101G prefix-XOR rank; gym 104768 C XOR-divisibility; Simplex LP solver.
Hard matrix problems: Fibonacci squared subarray sum with segment-tree 4-tuple and companion matrix; cycle-decomposition partition state space for random-swap sorting, first-step analysis, and Gaussian elimination.