Definition 1 (Reduction Algorithm). Given a square matrix A, we can apply the reduction algorithm to obtain PALU where U is an upper triangular matrix, Lis a lower triangular matrix with 1's on the diagonal, and P is a permutation matrix. Then det (P) det(A) = ±det(A) det (L) det (U)= det(U) det (U) = U11. Unn. = Definition 2 (Cofactor Formula). det(A) where Cij = (-1)+det(Mij and Mij is the submatrix of A with row i and column j removed. Definition . Σ sgn(o)a10(1)... ano (n). σESym[n] n Σ aij Cij j=1 = 3 (Leibniz Formula). det(A)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.