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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-lambda*I)=0. Eigenvectors: solve (A-lambda*I)v=0.
Rank-Nullity: rank(A)+nullity(A)=n. Dot product: u·v=|u||v|cos(theta). Orthogonal: u·v=0. |