What does a zero determinant mean?

The matrix is singular: it squashes space into a lower dimension, so some information is lost and it cannot be inverted. For a linear system, it means there is either no solution or infinitely many. Its columns are linearly dependent.

What are eigenvalues used for in practice?

They describe the natural modes of a system. Engineers use them for vibration frequencies and stability, data scientists for principal component analysis, and search ranking and Markov chain analysis rely on the dominant eigenvector. They also simplify repeated matrix powers.

What is the difference between the dot product and the cross product?

The dot product returns a single number that measures how much two vectors point the same way, and it is zero for perpendicular vectors. The cross product, defined in three dimensions, returns a new vector perpendicular to both, with length equal to the area of the parallelogram they span.

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Linear Algebra calculators

These calculators cover matrix arithmetic, determinants, inverses, transposes, row reduction, LU decomposition, eigenvalues, Gram-Schmidt orthogonalization and vector operations including dot, cross and projection. They show intermediate steps so you can see how each operation is carried out.

12 tools · Updated October 21, 2026

Showing 12 tools

a×bCross Product CalculatorVectors(1, 2, 3) × (4, 5, 6)(−3, 6, −3)

λEigenvalue & Eigenvector CalculatorSolving and decompositionEigenvalues of [4 1; 2 3]λ = 5, 2

u⊥vGram-Schmidt CalculatorSolving and decompositionOrthogonalize (3, 1), (2, 2)(3, 1), (−2/5, 6/5)

[ ]Matrix CalculatorMatrix operations[1 2 0; −1 3 2] × [2 1; 0 −1; 4 3][2 −1; 6 2]

|A|Matrix Determinant CalculatorMatrix operationsdet [2 −3 1; 2 0 −1; 1 4 5]49

A⁻¹Matrix Inverse CalculatorMatrix operationsInverse of [3 4; 1 2][1 −2; −1/2 3/2]

LUMatrix LU DecompositionSolving and decomposition[2 1 −1; −3 −1 2; −2 1 2]x = (8, −11, −3)x = (2, 3, −1)

ABMatrix Multiplication CalculatorMatrix operations[2 3 4; 1 0 0] × [0 1000; 1 100; 0 10][3 2340; 0 1000]

rrefMatrix RREF CalculatorSolving and decompositionRREF [1 −2 1 | 0; 0 2 −8 | 8; 5 0 −5 | 10]x₁ = 1, x₂ = 0, x₃ = −1

AᵀMatrix Transpose CalculatorMatrix operationsTranspose of [1 2 3; 4 5 6][1 4; 2 5; 3 6]

a+bVector CalculatorVectors(3, 1) + (1, 2)(4, 3)

projVector Projection CalculatorVectorsProject (7, 6) onto (4, 2)(8, 4)

Which one do I need?

I want to add, multiply or invert matrices in one placeMatrix Calculator I need the determinant of a square matrixMatrix Determinant Calculator I need the inverse of a matrix with the steps shownMatrix Inverse Calculator I want to row reduce a matrix to solve a system or find its rankMatrix RREF Calculator I need eigenvalues and eigenvectors of a 2×2 or 3×3 matrixEigenvalue & Eigenvector Calculator I need the cross product of two 3D vectorsCross Product Calculator I want to project one vector onto anotherVector Projection Calculator

Matrices as transformations

A matrix can be read as a transformation that stretches, rotates, reflects or shears space. Multiplying matrices composes transformations, which is why order matters: rotating then scaling is generally different from scaling then rotating, and AB usually does not equal BA.

The determinant measures how a transformation scales area or volume. A determinant of zero means space is flattened into a lower dimension, so the matrix has no inverse. The Matrix Determinant and Matrix Inverse calculators show the cofactor or row-reduction steps, and the Matrix Calculator is the quicker choice for routine arithmetic.

Solving systems: RREF and LU

Row reduction to reduced row echelon form is the most direct way to solve a linear system by hand and to find rank, pivot columns and free variables. The Matrix RREF Calculator records each row operation, which matches how most courses expect the work to be shown.

LU decomposition factors a matrix into lower and upper triangular parts. Once that is done, solving Ax = b for many different right-hand sides is fast, and the determinant comes almost for free. Use Matrix LU Decomposition when you need the factorization itself or repeated solves.

Eigenvalues, orthogonality and vectors

An eigenvector is a direction that a matrix only stretches, and its eigenvalue is the stretch factor. Eigenvalues determine stability in dynamic systems, principal components in data analysis and natural frequencies in vibration problems. The Eigenvalue Calculator handles 2×2 and 3×3 matrices, including complex eigenvalues.

Gram-Schmidt turns a set of independent vectors into an orthonormal basis, the basis of QR decomposition. For everyday vector work, the Vector Calculator gives dot products and magnitudes, the Cross Product Calculator gives a vector perpendicular to two others, and the Vector Projection Calculator splits a vector into parallel and perpendicular parts.

Questions people ask

Each matrix represents a transformation, and applying transformations in a different order usually produces a different result. In many cases BA is not even defined when AB is, because the dimensions do not line up. Special pairs, such as a matrix and its inverse, do commute.