Matrix Formulas & Reference

Linear Algebra for Machine Learning · Axiom Academy

Matrix Multiplication (Row × Column) 3×3 Determinant (Cofactor Expansion) det(D) = product of diagonal entries Property: Q⁻¹ = Qᵀ and det(Q) = ±1 Basis = pivot columns of original matrix Dimension = nullity(A) = n - rank(A) Distributive: A(B+C) = AB + AC NOT commutative: AB ≠ BA (in general) Transpose of sum: (A+B)ᵀ = Aᵀ + Bᵀ Transpose of product: (AB)ᵀ = BᵀAᵀ Invertibility: A is invertible iff det(A) ≠ 0 Rank constraint: rank(AB) ≤ min(rank(A), rank(B))

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