Linear Algebra
College · 3 topic explainersMatrix computation and the geometry of vectors.
Linear Algebra Calculators
Linear algebra is the mathematics of vectors and matrices, and it powers everything from solving systems of equations to computer graphics and data science. These calculators work with exact fractions, so the row operations, cofactor expansions and characteristic polynomials they show match what you would write by hand.
Calculators
Matrix Calculator
Multiply, add or subtract matrices, scale or raise one to a power, transpose it, row reduce it to RREF, or find its rank and trace, with exact fractions and every step.
Determinant Calculator
Enter a square matrix to find its determinant, using the ad − bc rule for 2×2, cofactor expansion for 3×3 and row reduction to triangular form for larger sizes, with exact arithmetic.
Inverse Matrix Calculator
Enter a square matrix to find its inverse with exact fractions, using the 2×2 shortcut or Gauss-Jordan elimination on [A | I], and see the check that A times its inverse is the identity.
Eigenvalue Calculator
Enter a square matrix to find its eigenvalues and eigenvectors, with the characteristic polynomial, exact roots including radicals and complex pairs, and the null-space calculation for every eigenvector.
Dot Product Calculator
Enter two vectors to get their dot product, magnitudes, the angle between them and the projection of one onto the other, with every multiplication and sum shown.
Cross Product Calculator
Enter two three-dimensional vectors to get their cross product by the determinant method, its magnitude as the area of the parallelogram they span, and a check that it is perpendicular to both.
Vector Calculator
Enter 2D or 3D vectors with whole numbers, decimals or fractions. Get the magnitude as an exact square root, the unit vector with rationalised components, sums, differences, scalar multiples and linear combinations, or x- and y-components from a magnitude and angle.
Topics covered
- Matrix operations
- Addition, multiplication, transpose, rank and row reduction.
- Determinants and inverses
- Cofactor expansion, Gauss–Jordan elimination and when a matrix is singular.
- Eigenvalues
- Characteristic polynomials, eigenvectors and multiplicities.
- Vectors
- Dot products, angles between vectors, projections and cross products.
Exact arithmetic for matrices
Matrix calculations go wrong easily when decimals are rounded at every step. These calculators use exact fractions throughout, so a row reduction or an inverse matrix is shown in the same form you would obtain by hand, and an entry such as never turns into . When a result such as an eigenvalue is irrational, the exact radical form is given together with a decimal approximation.
Each operation is shown one step at a time: the row operation applied, the cofactor expanded or the characteristic polynomial being factored. That makes the tools useful for checking homework, where a single sign error early in a calculation changes every later line.
Where linear algebra is used
Systems of linear equations, computer graphics, machine learning, economics and quantum physics are all built on vectors and matrices. A determinant tells you whether a system has a unique solution, eigenvalues describe the long-term behavior of a process, the dot product measures angles and projections, and the cross product gives a vector perpendicular to two others. Understanding these operations by hand makes software output much easier to interpret.
Courses and topics
Topic explainers with worked examples, common mistakes and practice questions in Linear Algebra.
Guides
- How to Check Your Math Homework and Answers
Almost every math answer can be checked without the answer key. Substituting back, working in reverse, estimating, testing special values and checking units catch the large majority of errors in a fraction of the time the original work took.