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
Probability & Statistics
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

Inverse Matrix & Solving Matrix Equations: Gauss-Jordan Interactive Demonstration

Coefficient Matrix A
Constant Vector b

Result

Row Operations

Basic Introduction

1) Inverse Matrix

If square matrix A is invertible, there exists A⁻¹ such that AA⁻¹ = A⁻¹A = I. The equivalent condition is det(A) ≠ 0.

2) Matrix Equation Ax=b

When A is invertible, the unique solution is x = A⁻¹b. If A is not invertible, there may be no solution or infinitely many solutions.

3) Gauss-Jordan Method

  • Apply row operations to [A|I]. When the left side becomes I, the right side is A⁻¹.
  • Apply row operations to [A|b] to determine the solution structure.