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Home › Math & Conversions › Core Math & Algebra › Linear Algebra Toolbox Linear Algebra Toolbox – Matrix Calculator & System Solver Enter a 2×2, 3×3, or 4×4 matrix to compute determinant, rank, trace, inverse (when it exists), and solve linear systems of the form Ax = b . All computations are numerical and performed locally in your browser using double-precision floating-point arithmetic. Matrix calculator & Ax = b solver Choose the matrix size, enter coefficients, and (optionally) a right-hand-side vector b . Then click Calculate to compute key linear algebra quantities. Matrix size (n × n) 2 × 2 3 × 3 4 × 4 Small matrices keep results readable and highlight core concepts. System vector b (optional) Leave all cells empty to skip solving Ax = b . Matrix A entries Empty cells are treated as 0. Use decimals with a dot (e.g., 1.5). Negative values are allowed. Calculate Clear inputs Numerical tolerance for rank/singularity: 1e-10 Determinant, rank, trace, inverse (if it exists), and system solution details will appear here after you click Calculate . What is linear algebra used for? Linear algebra is the mathematical framework for working with linear relationships. Vectors and matrices capture systems of equations, geometric transformations, and data in high dimensions. Behind many modern technologies—computer graphics, recommendation systems, machine learning, optimization, and control theory—there is a layer of linear algebra doing the heavy lifting. 1. Matrices and linear systems A system of linear equations in \(n\) variables can be written compactly as \[ A x = b, \] where \(A\) is an \(n \times n\) matrix of coefficients, \(x\) is the unknown vector, and \(b\) is the right-hand-side vector. Solving the system means finding all vectors \(x\) that satisfy the equation. If \(A\) is invertible, the solution is unique and can be written as \(x = A^{-1} b\). When \(A\) is singular or nearly singular, the solution may not exist or may not be unique. 2. Determinant, rank, and trace Determinant \(\det(A)\) measures how volumes scale under the linear transformation represented by \(A\). A zero determinant means the transformation collapses some dimension and \(A\) is singular (not invertible). Rank is the dimension of the image of \(A\); it tells you how many independent directions survive after applying the transformation. For an \(n \times n\) matrix, full rank means rank \(n\). Trace \(\operatorname{tr}(A)\) is the sum of diagonal elements of \(A\). For many matrices, it also equals the sum of eigenvalues (counted with multiplicity). 3. Inverse matrices and singularity A square matrix \(A\) is invertible if there exists another matrix \(A^{-1}\) such that \(A A^{-1} = A^{-1} A = I\), where \(I\) is the identity matrix. Numerically: If \(\det(A)\) is very close to zero, \(A\) is ill-conditioned: small changes in the data can cause large changes in the solution. If the rank of \(A\) is less than \(n\), then \(A\) is singular and has no inverse. The toolbox uses Gaussian elimination with partial pivoting to estimate rank, determinant, and, where stable, the inverse and system solution. 4. Eigenvalues and beyond Eigenvalues and eigenvectors describe directions that a linear map scales without rotating. They are fundamental in dynamical systems, principal component analysis (PCA), and spectral graph theory. For clarity and robustness, this introductory toolbox focuses on determinant, rank, trace, and solving small systems, which are the building blocks you will typically meet first in a linear algebra course. Linear algebra toolbox – FAQ How should I interpret the determinant being “very small” but not exactly zero? + If the determinant is tiny relative to the scale of the entries, the matrix is nearly singular . Numerically, this means solving Ax = b or computing A −1 will be sensitive to rounding and data errors. The toolbox flags such situations by using a small tolerance when deciding whether a pivot is effectively zero. What does it mean if the tool says that the system Ax = b is inconsistent? + An inconsistent system is one where no vector x satisfies all the equations simultaneously. In row-echelon form, this appears as a row of the form [0 0 … 0 | c] with c ≠ 0 in the augmented matrix [A | b]. In that case, the tool reports that the system has no solution under the given numeric tolerance. Why does the tool limit matrices to 4 × 4? + The focus of this page is clarity and reliability for hand-sized problems. For larger matrices, numerical stability, performance, and visualization require more sophisticated interfaces and back-end libraries. For study, exam preparation, or small engineering checks, 2×2–4×4 is typically enough to illustrate the main ideas. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox (you are here) Trapezoidal Rule Perfect Number Checker Chinese Remainder Theorem Set Theory Basics Geometry & analysis tools Pythagorean Theorem Convolution Explorer Dice Roll Probability Base Converter Interval Notation Helper Professional use note This linear algebra toolbox is ideal for education, exam practice, and quick engineering checks. For large-scale or safety-critical applications, verify key results with independent numerical software or formal analysis.
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