Enter matrix values
Use the responsive input grid below. Large screens show three columns, smaller screens show two, and mobile shows one.
Example data table
| Example | a11 | a12 | a13 | a21 | a22 | a23 | a31 | a32 | a33 | Determinant |
|---|---|---|---|---|---|---|---|---|---|---|
| Worked sample | 2 | -1 | 3 | 0 | 4 | 5 | 1 | 2 | -2 | -53 |
This sample produces positive diagonal sum -21 and negative diagonal sum 32, giving determinant -53.
Formula used
For a matrix A = [[a, b, c], [d, e, f], [g, h, i]], the 3x3 shortcut determinant formula is:
det(A) = aei + bfg + cdh − ceg − afh − bdi
This shortcut is Sarrus’ rule. Add the three forward diagonal products, add the three backward diagonal products, then subtract the backward total from the forward total.
How to use this calculator
- Enter the nine matrix values in the 3x3 grid.
- Choose how many decimal places you want in the outputs.
- Set the graph range and number of points for scalar scaling.
- Press Calculate determinant to show the result above the form.
- Review diagonal products, determinant, trace, and matrix classification.
- Use the download buttons to export the current result as CSV or PDF.
FAQs
1. What shortcut does this calculator use?
It uses Sarrus’ rule for 3x3 determinants. The method repeats the first two columns mentally, adds three forward diagonal products, adds three backward products, then subtracts the backward total.
2. Does this work for any 3x3 matrix?
Yes. The shortcut works for any numeric 3x3 matrix, including integers, decimals, negatives, and zeros. It does not apply directly to 2x2, 4x4, or larger matrices.
3. What does a zero determinant mean?
A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, and the matrix does not have an inverse.
4. Why does the calculator show positive and negative diagonals?
Those values let you verify the shortcut step by step. Seeing each product separately helps catch sign errors and makes classroom checking much easier.
5. What is the graph showing?
The graph shows how the determinant changes when the whole matrix is multiplied by a scalar k. For a 3x3 matrix, det(kA) equals k cubed times det(A).
6. Can I export my result?
Yes. After calculating, you can download a CSV for spreadsheet use or a PDF summary for sharing, printing, or keeping a record of your matrix work.
7. Why is trace included with the determinant?
Trace is not part of the determinant shortcut, but it is another useful matrix summary. Many students like seeing both values together during practice and review.
8. How can I avoid determinant mistakes?
Enter values carefully, keep negative signs visible, review both diagonal sums, and compare the expanded formula. Loading the example matrix is also helpful for checking your workflow.