metadata
source: knowledge_base
topic: linear_algebra
class_level: class_12
difficulty: advanced
Linear Algebra Essentials: Matrix multiplication: row-by-column. Determinant 2x2: ad-bc. Inverse: A^(-1)=adj(A)/det(A), exists iff det(A)≠0. Eigenvalues: det(A-lambdaI)=0. Eigenvectors: solve (A-lambdaI)v=0. Rank-Nullity: rank(A)+nullity(A)=n. Dot product: u·v=|u||v|cos(theta). Orthogonal: u·v=0.