Free Linear Algebra Calculators

Matrix Determinant Calculator

Calculate the determinant of 2×2 and 3×3 matrices.

Matrix Multiplication Calculator

Multiply two matrices together with step-by-step dot products.

Vector Calculator | Dot & Cross Product

Add, subtract, find dot product, cross product, and magnitude of vectors.

Matrix Inverse Calculator

Calculate the inverse of a 2×2 or 3×3 matrix with step-by-step elimination.

Eigenvalue & Eigenvector Calculator | 2×2 & 3×3

Calculate eigenvalues and eigenvectors for 2×2 and 3×3 matrices. Shows characteristic polynomial, eigenvectors, diagonalization, matrix properties, and step-by-step working.

Matrix Transpose Calculator

Transpose matrices of any size by swapping rows and columns.

Matrix RREF Calculator

Reduce any matrix to Row Reduced Echelon Form (RREF) using Gauss-Jordan elimination. Shows pivot positions, rank, and step-by-step row operations.

Matrix Calculator | Add, Multiply & Invert

Perform matrix addition, subtraction, multiplication, transposition, and find the determinant and inverse for matrices up to 4×4. Step-by-step working shown for all operations.

Cross Product Calculator | 3D Vector a × b

Calculate the cross product of two 3D vectors, find the resulting vector's magnitude and unit vector, verify orthogonality to both inputs, and compute the area of the parallelogram they span. Full determinant working shown with angle between vectors.

Matrix LU Decomposition | Solve Ax=b, Determinant & Step-by-Step

Decompose any 2×2, 3×3, or 4×4 matrix A into lower-triangular L and upper-triangular U using Doolittle's method. Shows each elimination step, uses LU to solve Ax=b via forward and back substitution, computes the determinant as the product of U's diagonal, and verifies A=LU.

Vector Projection Calculator | Projection, Decomposition & Gram-Schmidt

Compute the projection of vector u onto v in 2D or 3D, decompose u into parallel and perpendicular components, find the angle between vectors, and orthogonalize up to three vectors using the Gram-Schmidt process. Computes unit vectors, magnitudes, dot products, and scalar projections with full working.

Gram-Schmidt Calculator | Orthogonalization, Orthonormal Basis & QR Decomposition

Apply the Gram-Schmidt process to 2–4 linearly independent vectors in R²–R⁴ to produce an orthogonal basis and orthonormal basis. Displays the full QR decomposition Q·R = A and verifies orthogonality via dot products.