Linear Algebra Calculators

Last updated: October 2026

Linear System Solver

Solve a system of two linear equations in two unknowns (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) to find the x and y that satisfy both at once.

Solve a system of two linear equations (ax + by = e and cx + dy = f) instantly.

Solve a system of two linear equations in two unknowns using Cramer's rule or substitution.

Compute the inverse of a 2×2 matrix and automatically detect singular matrices where no inverse exists.

Determine the rank of any matrix up to 3×4 using Gaussian elimination.

Project vector u onto vector v in 2D or 3D: get the projection vector, the scalar projection or the projection's length.

Compute the determinant of a 2×2 matrix as a11·a22 − a12·a21.

Compute the dot product of two 3D vectors as v1·v2 = v1x·v2x + v1y·v2y + v1z·v2z.

Compute the angle in degrees between two 2D vectors using cos θ = (v1·v2)/(|v1|·|v2|) and the arccosine function.

Find the straight-line (Euclidean) distance between two points in a 2D plane using their x and y coordinates.

Compute the trace of a 3×3 matrix as the sum of its diagonal entries: a11 + a22 + a33.

Compute the scalar magnitude of the cross product of two 2D vectors U and V.

Find both eigenvalues of any 2×2 matrix using the characteristic equation.

Compute the eigenvalues and eigenvectors of a 2×2 matrix instantly.

Evaluate a quadratic form Q(x) = x^T A x for a symmetric 2×2 matrix and a vector (x₁, x₂).