About Matrix Calculator
This matrix calculator is free, needs no sign-up and runs in your browser, so the matrices you enter are never uploaded. Fill in a grid up to 6 × 6, choose an operation and read the result with the steps beside it. It covers addition, subtraction, multiplication, transpose, determinant, inverse, rank, reduced row echelon form, the adjugate with its cofactors, and eigenvalues.
Entries can be integers, decimals or fractions such as 3/4, and results are shown as exact fractions by default. A switch changes the output to decimals when you want to read the numbers more easily. Blank cells count as zero, so you only need to fill in the non-zero entries.
The working is typeset as matrices: the 2 × 2 determinant rule, the first-row expansion for 3 × 3, every row operation for bigger matrices and for inverses, and the characteristic polynomial for eigenvalues. A product can be opened to show the working of every entry. LU and QR decompositions are included too. Exact fractions mean no rounding error creeps into a long elimination, and the decimal switch changes the steps as well as the answer.
How to use Matrix Calculator
- Choose an operation, such as Determinant, Inverse or A × B.
- Set the size of matrix A with the row and column selectors. Square operations use a square grid, and two-matrix operations also show matrix B.
- Type the entries. Use integers, decimals or fractions like 3/4, and leave a cell blank for zero.
- The tool opens on a worked example. Read the result, typeset as a determinant, a rank or a matrix.
- Read the steps for the working, and use the decimal switch if you would rather see decimals than fractions.
Features
- Grids up to 6 × 6, with rows and columns set separately for non-square matrices.
- Addition, subtraction and multiplication of two matrices, with a clear message when the sizes do not match.
- Determinant with the 2 × 2 rule, first-row expansion for 3 × 3 and row reduction for larger matrices.
- Inverse by row-reducing [A | I], with a plain message when the matrix is singular.
- Rank and reduced row echelon form with each row operation listed.
- Adjugate and cofactor matrix for square matrices.
- Eigenvalues from the characteristic polynomial, including complex ones, with eigenvectors: exact for rational eigenvalues and numeric (labelled) for irrational ones.
- LU decomposition (with row swaps when a pivot is zero) and QR decomposition by Gram-Schmidt, exact with square roots.
- Fraction entries and fraction results, with a switch to decimals that also applies to the steps.
- Transpose of any matrix.
The core rules the calculator follows
The determinant of a 2 × 2 matrix [[a, b], [c, d]] is ad − bc. A square matrix has an inverse exactly when its determinant is not zero, and for 2 × 2 the inverse is 1 ÷ det times [[d, −b], [−c, a]]. A product AB exists only when A has as many columns as B has rows, and each entry is a row of A times a column of B.
Eigenvalues are the roots of the characteristic polynomial det(A − λI) = 0. Their sum equals the trace of A and their product equals det(A), which the calculator uses as a check.
det [[a, b], [c, d]] = ad − bc
A⁻¹ = (1 / det) × [[d, −b], [−c, a]]
(AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ
det(A − λI) = 0Worked examples
det [[1, 2], [3, 4]] gives −2
1 × 4 − 2 × 3 = 4 − 6 = −2.
[[1, 2, 3], [4, 5, 6]] × [[7, 8], [9, 10], [11, 12]] gives [[58, 64], [139, 154]]
A is 2 × 3 and B is 3 × 2, so the product is 2 × 2. Top-left: 1×7 + 2×9 + 3×11 = 58.
Inverse of [[4, 7], [2, 6]] gives [[3/5, −7/10], [−1/5, 2/5]]
det = 24 − 14 = 10. A⁻¹ = (1/10) × [[6, −7], [−2, 4]].
Eigenvalues of [[2, 1], [1, 2]] gives λ = 3 and λ = 1
The characteristic polynomial is λ² − 4λ + 3. The trace is 4 and the determinant is 3, which match the sum and product of the roots.
Which operations need which shape
| Operation | Shape needed | Result |
|---|---|---|
| A + B, A − B | A and B the same size | Same size |
| A × B | Columns of A = rows of B | Rows of A by columns of B |
| Transpose | Any | Columns by rows |
| Determinant | Square | A number |
| Inverse | Square, det ≠ 0 | Square matrix |
| Adjugate, eigenvalues | Square | Matrix, list of numbers |
| Rank, RREF | Any | A number, a matrix |
Good to know
When a matrix has no inverse
A square matrix with determinant zero is singular, and the calculator says so instead of returning a result. The rows are then linearly dependent: one row can be built from the others. [[1, 2], [2, 4]] is an example, since the second row is twice the first and the determinant is 1 × 4 − 2 × 2 = 0. Its rank is 1, less than the size 2, which is the same fact seen another way.
Rank and RREF
The rank is the number of pivot rows after row reduction. The matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] looks full but has rank 2, because its reduced form is [[1, 0, −1], [0, 1, 2], [0, 0, 0]]. The same reduced form is what you read to solve a linear system written as an augmented matrix.
What the eigenvalue mode does and does not do
It finds eigenvalues, the scalars λ where det(A − λI) = 0, and shows the characteristic polynomial. It does not list eigenvectors. For a real matrix with complex eigenvalues, such as the rotation [[0, −1], [1, 0]], the eigenvalues come out as i and −i.
Privacy: how it works
- All the arithmetic runs in your browser tab. Your matrices are not sent to Ulkaa or anyone else, and there is nothing to sign in to.
Tips
- Write fractions as 3/4 rather than 0.75 if you want results to stay exact.
- Leave entries blank for zero instead of typing every zero in a large grid.
- To solve a linear system, put the coefficients and constants in one grid and use RREF, or use the equation solver.
- Multiplication is not commutative. To check AB against BA, swap which matrix is A and which is B.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your matrices are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your matrices are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find a determinant?
For a 2 × 2 matrix [[a, b], [c, d]] it is ad − bc. A 3 × 3 is expanded along the first row into three 2 × 2 determinants, and larger ones are reduced with row operations.
When does an inverse not exist?
When the matrix is not square or its determinant is zero. Such a matrix is called singular.
How do I multiply matrices?
Take each row of A and each column of B, multiply matching entries and add them. The number of columns in A must equal the number of rows in B.
What is the rank of a matrix?
It is the number of linearly independent rows, which equals the number of pivots after row reduction.
How do I find eigenvalues?
Solve det(A − λI) = 0. The calculator expands this characteristic polynomial and solves it, so you see both the polynomial and its roots.
How large a matrix can I use?
Up to 6 × 6 for each matrix.
Can I enter fractions and see fractional answers?
Yes. Type entries such as 3/4 and results are exact fractions. Tick the decimal option to see decimals instead.
Popular Matrix Calculator pages
- Matrix Determinant CalculatorFind the determinant of any square matrix up to 6 × 6.
- Inverse Matrix CalculatorFind A⁻¹ for a square matrix, or learn why it does not exist.
- Matrix Multiplication CalculatorMultiply matrix A by matrix B.
- Matrix Addition CalculatorAdd or subtract two matrices of the same size.
- Matrix Transpose CalculatorSwap the rows and columns of a matrix.
- Matrix Rank CalculatorFind how many independent rows a matrix has.
- Eigenvalue CalculatorFind the eigenvalues of a square matrix and see the characteristic polynomial.
- 2x2 Matrix Determinant CalculatorEnter a, b, c and d to get ad − bc.
- 3x3 Matrix Determinant CalculatorExpand a 3 × 3 determinant along the first row with every minor shown.
- 3x3 Inverse Matrix CalculatorInvert a 3 × 3 matrix with every row operation listed.
- RREF CalculatorRow-reduce any matrix up to 6 × 6 to reduced row echelon form.
- Adjugate and Cofactor Matrix CalculatorGet the cofactor matrix, its transpose the adjugate, and the determinant.
- LU Decomposition CalculatorFactor a square matrix into a lower and an upper triangular matrix, with exact fractions.
- QR Decomposition CalculatorFactor a matrix into an orthonormal Q and an upper triangular R, in exact form.