Linear Algebra Calculators
Last updated: September 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.
2x2 Linear System Solver
Solve a system of two linear equations (ax + by = e and cx + dy = f) instantly.
2x2 System of Linear Equations Solver
Solve a system of two linear equations in two unknowns using Cramer's rule or substitution.
Vector Projection Calculator
Compute the vector projection of u onto v in 2D or 3D space, returning each component of the projected vector.
2x2 Matrix Inverse Calculator
Compute the inverse of a 2×2 matrix and automatically detect singular matrices where no inverse exists.
Matrix Rank Calculator
Determine the rank of any matrix up to 3×4 using Gaussian elimination.
Matrix Inverse Calculator
Find the inverse of a 2×2 matrix instantly, with a singular-matrix check.
Vector Magnitude Calculator
Compute the Euclidean magnitude (length) of a 3D vector as √(x² + y² + z²).
2x2 Matrix Determinant Calculator
Compute the determinant of a 2×2 matrix as a11·a22 − a12·a21.
Vector Dot Product Calculator
Compute the dot product of two 3D vectors as v1·v2 = v1x·v2x + v1y·v2y + v1z·v2z.
Distance Between Points Calculator
Find the straight-line (Euclidean) distance between two points in a 2D plane using their x and y coordinates.
3D Vector Cross Product Calculator
Calculate the cross product of two 3D vectors, returning a new vector perpendicular to both along with its magnitude.
Gram-Schmidt Orthogonalization
Convert a set of linearly independent 2D vectors into a mutually orthogonal (or orthonormal) basis using the Gram-Schmidt process.
Linear Transformation Calculator
Apply rotation, scaling, or reflection transformations to 2D vectors and see the resulting coordinates.
Vector Cross Product Calculator
Compute the cross product of two 3D vectors A and B, returning the resulting perpendicular vector.
3x3 Matrix Trace Calculator
Compute the trace of a 3×3 matrix as the sum of its diagonal entries: a11 + a22 + a33.
Angle Between Vectors Calculator
Compute the angle in degrees between two 2D vectors using cos θ = (v1·v2)/(|v1|·|v2|) and the arccosine function.
Matrix Scalar Multiplication
Multiply every element of a 2×2 matrix by a single scalar value.
2D Vector Cross Product
Compute the scalar magnitude of the cross product of two 2D vectors U and V.
Gram-Schmidt Process Calculator
Transform a set of linearly independent vectors into an orthogonal or orthonormal basis using the Gram-Schmidt process.
QR Decomposition Calculator
Decompose any matrix A into an orthogonal matrix Q and an upper-triangular matrix R such that A = QR.
2x2 Matrix Eigenvalue Calculator
Find both eigenvalues of any 2×2 matrix using the characteristic equation.
Linear Transformation Matrix Calculator
Apply 2D linear transformations — rotation, scaling, reflection, and shearing — to any input vector.
Eigenvalue Calculator
Compute the eigenvalues and eigenvectors of a 2×2 matrix instantly.
Quadratic Form Calculator
Evaluate a quadratic form Q(x) = x^T A x for a symmetric 2×2 matrix and a vector (x₁, x₂).
Matrix Trace and Properties Calculator
Compute the trace, Frobenius norm, spectral radius, and other key properties of a matrix in one place.