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).

Outils associés

∑ Mean, Median & Mode Compute mean, median & mode ∑ Standard Deviation Calculate population & sample SD ∑ Variance Calculator Calculate population & sample var ∑ Correlation Coefficient Pearson/Spearman correlation 📈 Regression Analysis Linear regression & prediction 🎲 Probability Calculator Independent/conditional probability ∑ Bayes' Theorem Bayesian posterior probability 🎲 Permutations & Combinations Calculate C(n,r) and P(n,r) 📐 Matrix Addition Matrix addition & subtraction 📐 Matrix Multiplication Matrix multiplication operation 📐 Matrix Determinant Compute square matrix determinant 📐 Matrix Inverse Compute invertible matrix inverse