linear algebra Calculators
2x2 Matrix Determinant Calculator
Instantly find the determinant of any 2×2 matrix by entering its four elements. Use it when solving systems of linear equations, checking matrix invertibility, or working through linear algebra homework.
Vector Dot Product Calculator
Compute the dot product of two 3D vectors by entering their X, Y, and Z components. Commonly used in physics, computer graphics, and engineering to find the angle between vectors or project one vector onto another.
Vector Magnitude Calculator
Find the length (magnitude) of a 3D vector by entering its X, Y, and Z components. Essential for normalizing vectors, computing distances from the origin, and working with physics or graphics problems.
3x3 Matrix Trace Calculator
Calculate the trace of a 3×3 matrix by summing its three main diagonal elements. Used in linear algebra, quantum mechanics, and machine learning when analyzing matrix properties like eigenvalue sums.
Angle Between Vectors Calculator
Find the angle in degrees between two 2D vectors given their X and Y components. Useful in physics, game development, and navigation when you need to know how far apart two directions are.
2x2 Linear System Solver
Solve a system of two linear equations (ax + by = e and cx + dy = f) instantly. Use it when you need exact values of x and y that satisfy both equations simultaneously.
Vector Projection Calculator
Calculate the scalar projection of 2D vector A onto vector B. Use this when you need to find how much of vector A lies in the direction of vector B.
Matrix Scalar Multiplication
Multiply every element of a 2×2 matrix by a single scalar value. Use this when scaling a transformation matrix or adjusting magnitudes in linear algebra problems.
Distance Between Points Calculator
Find the straight-line (Euclidean) distance between two points in a 2D plane using their x and y coordinates. Ideal for geometry homework, mapping, and coordinate-based programming tasks.
2D Vector Cross Product
Compute the scalar magnitude of the cross product of two 2D vectors U and V. Use it to find the signed area of the parallelogram they span or to test if two vectors are parallel.
Matrix Determinant Calculator
Compute the determinant of any 2×2, 3×3, or 4×4 square matrix instantly. Use it to check invertibility, solve linear systems, or evaluate transformations in linear algebra.
2x2 Matrix Eigenvalue Calculator
Find both eigenvalues of any 2×2 matrix using the characteristic equation. Essential for stability analysis, principal component analysis, and differential equations.
2x2 Matrix Inverse Calculator
Compute the inverse of a 2×2 matrix and automatically detect singular matrices where no inverse exists. Useful for solving linear systems, transformations, and cryptography.
3D Vector Cross Product Calculator
Calculate the cross product of two 3D vectors, returning a new vector perpendicular to both along with its magnitude. Used in physics, engineering, and 3D graphics.
2x2 System of Linear Equations Solver
Solve a system of two linear equations in two unknowns using Cramer's rule or substitution. Ideal for algebra homework, engineering problems, and quick verification of hand calculations.
Matrix Rank Calculator
Determine the rank of any matrix up to 3×4 using Gaussian elimination. Use it when solving linear systems, checking linear independence, or verifying that a matrix has full rank.
Quadratic Form Calculator
Evaluate a 2-variable quadratic form Q(x₁, x₂) and analyze its definiteness. Useful in optimization, geometry, and studying conic sections.
Gram-Schmidt Orthogonalization Calculator
Transform a set of linearly independent vectors into an orthonormal basis using the Gram-Schmidt process. Essential for QR decomposition, least-squares problems, and numerical linear algebra.
Linear Transformation Matrix Calculator
Apply 2D linear transformations — rotation, scaling, reflection, and shearing — to any input vector. Ideal for computer graphics, robotics, and linear algebra coursework.
Eigenvalue Calculator
Compute the eigenvalues and eigenvectors of a 2×2 matrix instantly. Used in stability analysis, principal component analysis, and quantum mechanics.
Linear System Solver
Solve systems of two or more linear equations simultaneously to find variable values that satisfy all equations at once. Use this when balancing chemical equations, solving circuit problems, or handling any multi-variable algebra.
Gram-Schmidt Process Calculator
Transform a set of linearly independent vectors into an orthogonal or orthonormal basis using the Gram-Schmidt process. Useful in linear algebra courses, QR decomposition, and signal processing applications.
Matrix Inverse Calculator
Find the inverse of a 2×2 or 3×3 matrix instantly. Use it when solving linear systems, transforming coordinates, or computing solutions in engineering and computer graphics.
QR Decomposition Calculator
Decompose any matrix A into an orthogonal matrix Q and an upper-triangular matrix R such that A = QR. Essential for solving least-squares problems, computing eigenvalues, and numerical linear algebra.
Matrix Trace and Properties Calculator
Compute the trace, Frobenius norm, spectral radius, and other key properties of a matrix in one place. Useful for checking matrix health, eigenvalue bounds, and verifying theoretical results in linear algebra.
Matrix Inverse Calculator
Find the inverse of a 2×2 or 3×3 matrix instantly. Use this when solving systems of linear equations, transforming coordinates, or working through linear algebra coursework.
Linear Transformation Calculator
Apply rotation, scaling, or reflection transformations to 2D vectors and see the resulting coordinates. Ideal for linear algebra students and graphics programmers exploring how matrices act on vectors.
Gram-Schmidt Orthogonalization
Convert a set of linearly independent 2D vectors into a mutually orthogonal (or orthonormal) basis using the Gram-Schmidt process. Used in linear algebra, QR decomposition, and numerical methods courses.
Vector Projection Calculator
Compute the vector projection of u onto v in 2D or 3D space, returning each component of the projected vector. Useful in physics, computer graphics, and linear algebra when decomposing forces or vectors along a given direction.
Gram-Schmidt Orthogonalization Calculator
Convert two linearly independent 2D vectors into an orthonormal basis using the Gram-Schmidt process. Ideal for linear algebra students working on QR decomposition, subspace projections, or eigenvalue problems.
Matrix Rank Calculator
Find the rank of a 2×2 matrix instantly using row reduction. Use this when solving systems of equations, checking linear independence, or determining if a matrix is invertible.
Vector Cross Product Calculator
Compute the cross product of two 3D vectors A and B, returning the resulting perpendicular vector. Ideal for physics, engineering, and graphics problems involving torque, normals, or rotational axes.
Quadratic Form Calculator
Evaluate a quadratic form Q(x) = x^T A x for a symmetric 2×2 matrix and a vector (x₁, x₂). Use it to test whether a matrix is positive definite, negative definite, or indefinite.