About the Matrix Calculator
What is the Matrix Calculator?
A matrix is a rectangular grid of numbers. Square matrices are used to solve systems of equations, rotate and scale computer graphics, describe networks and transitions, and model economies. The most important single number of a square matrix is its determinant.
Enter a 3×3 matrix (or use its top-left 2×2 block) and choose what the detailed table should show: the inverse, the transpose, the adjugate or A × A. The result card always shows the determinant, the trace and whether the matrix can be inverted.
How to use the Matrix Calculator
- Matrix Size (sample: 3 × 3)
- Show in Table — the matrix shown in the detailed table below the result. (sample: Inverse matrix)
- Row 1, Column 1 (sample: 2)
- Row 1, Column 2 (sample: 1)
- Row 1, Column 3 (sample: 1)
- Row 2, Column 1 (sample: 1)
- Row 2, Column 2 (sample: 3)
- Row 2, Column 3 (sample: 2)
- Row 3, Column 1 (sample: 1)
- Row 3, Column 2 (sample: 0)
- Row 3, Column 3 (sample: 0)
- Press Calculate (results also refresh as you type). The result card shows the Determinant, along with Trace (sum of diagonal), Invertible? and Table shows.
Formula used
A⁻¹ = adj(A) ÷ det(A)
- det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
- adj(A) = the transpose of the matrix of cofactors
For a 2×2 matrix [[a, b], [c, d]], det = ad − bc and the inverse is [[d, −b], [−c, a]] ÷ (ad − bc). A matrix with determinant 0 is singular and has no inverse.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Matrix Size | 3 × 3 |
| Show in Table | Inverse matrix |
| Row 1, Column 1 | 2 |
| Row 1, Column 2 | 1 |
| Row 1, Column 3 | 1 |
| Row 2, Column 1 | 1 |
| Row 2, Column 2 | 3 |
| Row 2, Column 3 | 2 |
| Result | Value |
|---|---|
| Determinant | -1 |
| Trace (sum of diagonal) | 5 |
| Invertible? | Yes |
| Table shows | Inverse |
Tips for using the Matrix Calculator
- Check an inverse by multiplying: A × A⁻¹ should give the identity matrix.
- Swapping two rows changes the sign of the determinant, and a row of zeros makes it 0.
- The trace equals the sum of the eigenvalues, and the determinant equals their product.