CYBERTOOLS / 0x42
LINEAR ALGEBRA RUNTIME // VECTOR SPACE OPERATOR

MATRIX CALCULATOR // ALGEBRA HUD

Real-time 2×2 and 3×3 matrix arithmetic: addition, multiplication, determinants, transposition, and matrix inversion.

MATRIX A
det(A) = 0
SELECT ALGEBRAIC OPERATION
MATRIX B
det(B) = 0
RESULT MATRIX // A + B
COMPUTATION TRACE & FORMULA IEEE 754 FLOAT
Select an operation to view linear algebra breakdown.

MATRIX CALCULATOR // LINEAR ALGEBRA FOUNDATIONS & MATRIX THEORY

Mathematical fundamentals of 2×2 and 3×3 matrix transformations, determinant geometry, and computational inversion.

Determinants & Geometric Scaling Factors

The determinant of a square matrix represents the volume scaling factor of the linear transformation described by the matrix.

  • 2×2 Determinant: Computed via the cross-multiplication formula: det(A) = ad − bc for matrix [[a,b],[c,d]]. Represents the signed area of the parallelogram formed by column vectors.
  • 3×3 Rule of Sarrus / Cofactor Expansion: Computed by expanding across minors along the top row: a(ei − fh) − b(di − fg) + c(dh − eg).
  • Singularity Condition: When det(A) = 0, the transformation collapses spatial dimensions into a lower-dimensional line or plane; consequently, the matrix is singular and possesses no multiplicative inverse.

Matrix Inversion & Cramer's Framework

For any non-singular square matrix A, its inverse A⁻¹ satisfies the fundamental equality A × A⁻¹ = I, where I is the identity matrix.

  • Adjugate Formulation: A⁻¹ = (1 / det(A)) × adj(A), where adj(A) is the transpose of the matrix of cofactors.
  • Computer Graphics & 3D Vectors: 3×3 and 4×4 matrices are the fundamental building blocks of 3D graphics rendering, Euler rotations, affine translation, and perspective projection pipelines.

Airgap Zero-Log Privacy Architecture

The Matrix Operator computes all algebraic operations locally inside browser memory using client-side JavaScript. No matrices, coordinate vectors, or calculation inputs are stored in cookies or transmitted across external networks.