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Module 3: Quick Reference

Determinants -- Linear Algebra

Formulas

2x2: det([a b;c d]) = ad - bc
nxn: cofactor expansion: det(A) = sum a_{ij}*(-1)^{i+j}*M_{ij} along any row/col
Triangular: det = product of diagonal entries

Row Operations

Swap

Negates det

Scale by c

Multiplies det by c

Row replacement

No change to det

Properties

det(AB)=det(A)det(B) | det(A^T)=det(A) | det(A^{-1})=1/det(A) | det(cA)=c^n det(A)

det=0 iff singular. Zero row/col or repeated rows forces det=0.

Cramer's Rule

x_i = det(A_i)/det(A), A_i = A with col i replaced by b

Geometry

2D

|det| = area of parallelogram

3D

|det| = volume of parallelepiped

Sign = orientation. det=0 = degenerate (dependent columns).