• i. Matrices and Matrix Power: Matrix product and powers, many power queries, geometric sums, semiring products ($(\min ,+)$, bottleneck, boolean), CSES Graph Paths and Strassen.
• ii. Gaussian Elimination, Rank and Inverse: Gaussian and Gauss–Jordan elimination over reals, ${\mathbb{F}}_p$ and ${\mathbb{F}}_2$: solution counts and kernels, rank uses, inverse, Lights Out, matching parity.
• iii. Determinants and Permanents: Determinants mod primes, composite moduli and exact (Bareiss), Lindström–Gessel–Viennot, permanent (Ryser) and hafnian (Björklund).
• iv. Characteristic Polynomial: Hessenberg $O(n^3)$ and Faddeev–LeVerrier, Cayley–Hamilton for $A^{k}v$, and $\det (A+xB)$ as a polynomial.
• v. Sparse and Structured Matrices: Wiedemann solve and determinant, circulant/Vandermonde/tridiagonal/rank-one formulas, Sherman–Morrison, Lanczos and Krylov methods.