linear algebra Calculators
Last updated: May 2026
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.
Vector Magnitude Calculator
Compute the Euclidean magnitude (length) of a 3D vector as √(x² + y² + z²).
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.
2x2 Linear System Solver
Solve a system of two linear equations (ax + by = e and cx + dy = f) instantly.
Vector Projection Calculator
Calculate the scalar projection of 2D vector A onto vector B.
Matrix Scalar Multiplication
Multiply every element of a 2×2 matrix by a single scalar value.
Distance Between Points Calculator
Find the straight-line (Euclidean) distance between two points in a 2D plane using their x and y coordinates.
2D Vector Cross Product
Compute the scalar magnitude of the cross product of two 2D vectors U and V.
2x2 Matrix Determinant Calculator
Compute the determinant of any 2×2, 3×3, or 4×4 square matrix instantly.
2x2 Matrix Eigenvalue Calculator
Find both eigenvalues of any 2×2 matrix using the characteristic equation.
2x2 Matrix Inverse Calculator
Compute the inverse of a 2×2 matrix and automatically detect singular matrices where no inverse exists.
3D Vector Cross Product Calculator
Calculate the cross product of two 3D vectors, returning a new vector perpendicular to both along with its magnitude.
2x2 System of Linear Equations Solver
Solve a system of two linear equations in two unknowns using Cramer's rule or substitution.
Matrix Rank Calculator
Determine the rank of any matrix up to 3×4 using Gaussian elimination.
Quadratic Form Calculator
Evaluate a 2-variable quadratic form Q(x₁, x₂) and analyze its definiteness.
Gram-Schmidt Orthogonalization Calculator
Transform a set of linearly independent vectors into an orthonormal basis using the Gram-Schmidt process.
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.
Linear System Solver
Solve systems of two or more linear equations simultaneously to find variable values that satisfy all equations at once.
Gram-Schmidt Process Calculator
Transform a set of linearly independent vectors into an orthogonal or orthonormal basis using the Gram-Schmidt process.
Matrix Inverse Calculator
Find the inverse of a 2×2 or 3×3 matrix instantly.
QR Decomposition Calculator
Decompose any matrix A into an orthogonal matrix Q and an upper-triangular matrix R such that A = QR.
Matrix Trace and Properties Calculator
Compute the trace, Frobenius norm, spectral radius, and other key properties of a matrix in one place.
Matrix Inverse Calculator
Find the inverse of a 2×2 or 3×3 matrix instantly.
Linear Transformation Calculator
Apply rotation, scaling, or reflection transformations to 2D vectors and see the resulting coordinates.
Gram-Schmidt Orthogonalization
Convert a set of linearly independent 2D vectors into a mutually orthogonal (or orthonormal) basis using the Gram-Schmidt process.
Vector Projection Calculator
Compute the vector projection of u onto v in 2D or 3D space, returning each component of the projected vector.
Gram-Schmidt Orthogonalization Calculator
Convert two linearly independent 2D vectors into an orthonormal basis using the Gram-Schmidt process.
Matrix Rank Calculator
Find the rank of a 2×2 matrix instantly using row reduction.
Vector Cross Product Calculator
Compute the cross product of two 3D vectors A and B, returning the resulting perpendicular vector.
Quadratic Form Calculator
Evaluate a quadratic form Q(x) = x^T A x for a symmetric 2×2 matrix and a vector (x₁, x₂).