About 3x3 Matrix Determinant Calculator
This calculator opens a 3 × 3 grid and finds the determinant by expanding along the first row. For [[2, 1, 3], [0, −1, 4], [1, 2, 5]] the three minors are −13, −4 and 1, and the determinant is −19. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The expansion is a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃, where each M is the 2 × 2 determinant left after removing that entry's row and column. Here the terms are +2 × (−13) = −26, −1 × (−4) = +4 and +3 × 1 = +3, and −26 + 4 + 3 = −19. The steps display each minor as a small matrix.
A second example is a singular one. For [[2, 0, 1], [1, 3, 2], [1, 1, 1]] the terms are 2, 0 and −2, which add to 0, so there is no inverse. The calculator accepts fractions and decimals in any cell.
How to use 3x3 Matrix Determinant 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.
First-row expansion
Alternate the signs +, −, + and multiply each first-row entry by its 2 × 2 minor.
det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)Worked examples
[[2, 1, 3], [0, −1, 4], [1, 2, 5]] gives −19
Minors: M₁₁ = (−1)(5) − (4)(2) = −13, M₁₂ = (0)(5) − (4)(1) = −4, M₁₃ = (0)(2) − (−1)(1) = 1. Terms: −26, +4, +3.
[[2, 0, 1], [1, 3, 2], [1, 1, 1]] gives 0
Terms 2, 0 and −2 sum to zero, so the matrix is singular.
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 the determinant of a 3x3 matrix?
Multiply each entry in the first row by the determinant of the 2 × 2 matrix left when you cross out its row and column, alternating the signs plus, minus, plus, then add.
Can I expand along a different row or column?
Yes, any row or column gives the same value if you keep the checkerboard sign pattern. The calculator always uses the first row.
More Matrix Calculator pages
- Matrix CalculatorAdd, multiply, invert, decompose (LU, QR) and analyse matrices up to 6 × 6, with typeset working.
- 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 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.