Math Visualization List
Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
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Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation

Multi-Row Expansion of Determinants: 2-Row Laplace Expansion Demo

Step-by-Step Process (Column Combinations C(n,2))

Term Column Combo Sign 2x2 Minor Cofactor Term Value Cumulative Sum

Basic Introduction

Multi-row expansion is a generalization of Laplace expansion. Select k rows (fixed as k=2 on this page), then enumerate the same number of column combinations.

Each term has the form:

(-1)^(sum of row indices + sum of column indices) × minor × cofactor

Here "cofactor" refers to the minor determinant obtained by removing the selected rows and columns.

Applications

  • For certain structured matrices, multi-row expansion is more efficient than single-row expansion.
  • It is a common tool for studying block determinants, Vandermonde structures, and algebraic identities.