Matrix Operations is a free online matrix calculator that shows every step of the solution, not just the answer. Pick one of the 25 operations below, enter your matrix, and get a worked solution you can follow line by line. It handles decimals, fractions, complex numbers and variables in matrices up to 20×20, and needs no sign-up.

All matrix operations

Arithmetic

Matrix addition calculatorAdd two matrices of the same size element by element.
Matrix subtraction calculatorSubtract one matrix from another of the same size, entry by entry.
Multiplying a matrix by a number calculatorMultiply every entry of a matrix by a number.
Matrix multiplication calculatorMultiply two matrices with the row-by-column rule, each entry worked out.
Power of a matrix calculatorRaise a square matrix to a whole-number power by repeated multiplication.

Properties

Matrix determinant calculatorFind the determinant by cofactor expansion, the Sarrus rule, row reduction or the Bareiss algorithm.
Matrix inverse calculatorInvert a square matrix with Gauss-Jordan elimination, the adjugate formula or the Bareiss algorithm.
Matrix minor calculatorCompute the minor of any entry: the determinant left after deleting its row and column.
Matrix cofactor calculatorCompute signed minors and build the full cofactor matrix.
Adjugate matrix calculatorBuild the adjugate matrix, the transpose of the cofactor matrix.
Rank of matrix calculatorFind the rank by row reduction or by the bordering-minors method.

Transformations

Transpose matrix calculatorSwap the rows and columns of any matrix.
Conjugate matrix calculatorReplace every complex entry with its complex conjugate.
Square root calculatorFind a matrix whose square is the matrix you entered.

Equations

Matrix equations calculatorSolve AX = B or XA = B for the unknown matrix X.
System of linear equations calculatorSolve a linear system by Gaussian elimination, Gauss-Jordan, Cramer's rule or the inverse matrix.

Eigen

Eigenvalues calculatorBuild the characteristic polynomial and find all its roots.
Eigenvectors calculatorFind the eigenvectors that belong to each eigenvalue.

Decompositions

LU decomposition calculatorFactor a square matrix into a lower and an upper triangular matrix.
QR decomposition calculatorFactor a matrix into an orthogonal Q and an upper triangular R by Gram-Schmidt, Householder or Givens.
Cholesky decomposition calculatorFactor a symmetric positive-definite matrix as L·Lᵀ.
Schur Decomposition calculatorFactor a square matrix as Q·T·Q* with unitary Q and upper triangular T.
Eigendecomposition(matrix diagonalization) calculatorDiagonalize a matrix as P·D·P⁻¹ from its eigenvalues and eigenvectors.
Polar decomposition calculatorFactor a matrix into a unitary matrix and a positive semi-definite matrix.
Singular Value Decomposition calculatorCompute the singular value decomposition A = U·Σ·Vᵀ.

What is a matrix?

A matrix is a rectangular table of numbers arranged in rows and columns. A matrix with m rows and n columns has size m×n, and the number in row i and column j is written aᵢⱼ. Matrices of the same size can be added entry by entry, and two matrices can be multiplied when the first has as many columns as the second has rows. Swapping rows and columns gives the transpose.

A square matrix has one number that says a lot about it, the determinant. When the determinant is not zero the matrix has an inverse, and the rank equals the size of the matrix. That is exactly the case in which a system of linear equations with this coefficient matrix has a single solution.

Matrices also describe linear transformations such as rotations, scalings and projections. The directions a transformation only stretches are its eigenvectors, and the stretch factors are its eigenvalues. Factorizations such as the LU, QR and singular value decompositions break a matrix into simpler pieces that make large computations faster and more stable.

Matrix Operations on your phone

The same calculator is available as a free app, with the same step-by-step solutions and offline use.

Google Play App Store