Matrix calculator

A matrix is a rectangle of numbers, and matrix operations are the language of linear algebra — solving systems of equations, transforming graphics, running regressions, propagating a neural network's weights.

Multiplication is not commutative

AB and BA are generally different matrices, and one may not even be defined. Multiplication requires the columns of A to match the rows of B: an m×n times an n×p gives an m×p. Each entry is the dot product of a row of A with a column of B.

(AB)ᵢⱼ = Σₖ Aᵢₖ × Bₖⱼ

The determinant

For a square matrix, the determinant is the factor by which the transformation scales area (2D) or volume (3D). A determinant of 0 means the transformation flattens space onto a lower dimension — information is destroyed, so it cannot be undone, which is exactly why singular matrices have no inverse.

Reduced row echelon form

RREF is the canonical simplified form: leading 1s stepping right and down, zeros everywhere else in those columns. It solves linear systems, reveals the rank, and finds a basis for the column space — all from one elimination.

Numerical notes

Elimination uses partial pivoting to stay stable, and values within 10⁻¹² of zero are displayed as 0 rather than as floating-point dust. For ill-conditioned matrices, treat the last couple of digits with suspicion.