How to use it
- Choose the operation, for example Determinant.
- Set the number of rows and columns.
- Fill in the matrix (whole numbers, decimals or fractions).
- Read the result and expand the elimination steps.
Operations
| Operation | Needs | Result |
|---|---|---|
| Determinant det(A) | Square A | A number, with steps |
| Inverse A⁻¹ | Square A with det ≠ 0 | A matrix, Gauss–Jordan steps |
| Multiply A · B | Columns of A = rows of B | A matrix |
| Add / subtract A ± B | Same size | A matrix |
| Scalar multiple k · A | Any A | A matrix |
| Transpose Aᵀ | Any A | Rows become columns |
| Rank | Any A | Number of pivots, with steps |
| Row echelon form | Any A | The eliminated matrix, with steps |
The calculator works with exact fractions, so 1/3 stays 1/3 instead of 0.333…, and there's no rounding error. If a result has fractions, tick Show as decimals to see them to 4 places.
Determinant: worked example
For A = [[6, 1, 1], [4, −2, 5], [2, 8, 7]] (the default example), expansion along the first row gives
det(A) = 6·(−2·7 − 5·8) − 1·(4·7 − 5·2) + 1·(4·8 − (−2)·2) = 6·(−54) − 18 + 36 = −306.
The calculator gets there by Gaussian elimination instead: it reduces A to an upper triangular matrix with row operations, multiplies the diagonal, and flips the sign for each row swap. Each step shows the row operation, for example R2 ← R2 − (2/3) · R1, and the matrix after it. If a zero row appears, the determinant is 0.
Inverse matrix
The inverse is found with Gauss–Jordan elimination on the augmented matrix [A | I]: row operations turn the left side into the identity matrix, and the right side becomes A⁻¹. If the left side can't be reduced to the identity (det A = 0), the matrix is singular and has no inverse; the calculator says so and shows where elimination stopped.
Check: A · A⁻¹ = I. You can verify it by choosing Multiply and pasting the result in as B.
Entering matrices
- Choose 1 to 6 rows and columns for each matrix.
- Entries can be whole numbers (
-3), decimals (2.5) or fractions (3/4). An empty cell counts as 0. - Matrix B appears only for operations that need two matrices.
To solve a system of linear equations, the system of equations solver sets up the augmented matrix for you and reports unique, infinite or no solutions.
Frequently asked questions
How do I find the determinant of a 3x3 matrix?
Choose Determinant, set 3 rows and 3 columns, and enter the entries. The calculator shows the elimination steps and the result.
When does a matrix have no inverse?
When its determinant is 0 (it's singular). Only square matrices can have an inverse.
Can I multiply matrices of different sizes?
Yes, if the number of columns of A equals the number of rows of B. A 2×3 times a 3×4 gives a 2×4 matrix.
Does it use fractions or decimals?
Exact fractions. You can switch the displayed result to decimals.
What is the rank of a matrix?
The number of nonzero rows (pivots) in its row echelon form, the number of linearly independent rows or columns.
How large can the matrices be?
Up to 6 rows and 6 columns.
Last updated: November 12, 2026