Math
Matrix Determinant and Inverse Calculator
Find the determinant, trace and inverse of a 2 × 2 or 3 × 3 matrix.
Determinant
18
- Trace (sum of the diagonal)
- 9
- Inverse
- Shown in the table
The determinant is 18, so the matrix has an inverse.
How it's worked out
- Determinant
expansion along the first row18
Inverse matrix
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| 0.6111 | -0.2222 | 0.0556 |
| -0.2222 | 0.4444 | -0.1111 |
| 0.0556 | -0.1111 | 0.2778 |
The formulas
2 × 2: det = ad − bc, inverse = (1 ÷ det) × [[d, −b], [−c, a]] · 3 × 3: inverse = adjugate ÷ det
Worked example
For [[2, 1, 0], [1, 3, 1], [0, 1, 4]], the determinant is 2(12 − 1) − 1(4 − 0) + 0 = 18, so the inverse exists.
Questions people ask
How do I find a 2 × 2 determinant?
For [[a, b], [c, d]], it's ad − bc.
When does a matrix have no inverse?
When its determinant is 0. The matrix is then called singular.
How is a 3 × 3 inverse found?
Each element's cofactor is worked out, the result is transposed (the adjugate) and divided by the determinant.
About this calculator
Formula and rules last checked 25 October 2026Written and tested by the CalculatorDex teamReport a problem