Linear Algebra Guide

Matrix

A rectangular array of numbers. An m×n matrix has m rows and n columns. Operations: addition, scalar multiplication, matrix multiplication (requires inner dimensions to match).

Determinant

For 2×2: det([[a,b],[c,d]]) = ad − bc. Non-zero determinant means the matrix is invertible. det = 0 means singular (no unique solution).

Inverse Matrix

A⁻¹ exists only if det(A) ≠ 0. A·A⁻¹ = I (identity matrix). For 2×2: A⁻¹ = (1/det)·[[d,-b],[-c,a]]

Eigenvalues & Eigenvectors

Av = λv, where λ is an eigenvalue and v is the eigenvector. Found by solving det(A − λI) = 0. Used in PCA, stability analysis, quantum mechanics.

Rank

Number of linearly independent rows (or columns). rank(A) = n means full rank (invertible for square). rank < n means dependent rows/columns exist.

Dot Product vs Matrix Multiply

Dot product: a·b = Σaᵢbᵢ (scalar result). Matrix multiply: C = AB where Cᵢⱼ = Σ Aᵢₖ Bₖⱼ (matrix result, requires A cols = B rows).