Matrix Determinant and Inverse Calculator (3×3)
Enter the nine entries of the matrix row by row (use 0 and 1 to embed a 2×2 matrix).
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Matrix Determinant and Inverse Calculator (3×3) by TheCalcTools
How the Matrix Determinant and Inverse Calculator (3×3) works
The determinant is found by cofactor expansion along the first row: det = m₁₁C₁₁ + m₁₂C₁₂ + m₁₃C₁₃, where each cofactor is a signed 2×2 determinant.
The inverse is the transpose of the cofactor matrix (the adjugate) divided by the determinant. A matrix with determinant 0 is singular and has no inverse.
The formula
Worked example
Using the default values shown in the calculator:
| Row 1, column 1 | 2 |
|---|---|
| Row 1, column 2 | -1 |
| Row 1, column 3 | 0 |
| Row 2, column 1 | 1 |
| Row 2, column 2 | 3 |
| Row 2, column 3 | 2 |
| Row 3, column 1 | 0 |
| Row 3, column 2 | 1 |
| Row 3, column 3 | 4 |
- For [[2, −1, 0], [1, 3, 2], [0, 1, 4]] the determinant is 24.000000.
- Inverse row 1: [ 0.416667, 0.166667, -0.083333 ].
Result:
| Determinant | 24.0000 |
|---|---|
| Trace | 9.0000 |
| Inverse, row 1 | [ 0.416667, 0.166667, -0.083333 ] |
| Inverse, row 2 | [ -0.166667, 0.333333, -0.166667 ] |
| Inverse, row 3 | [ 0.041667, -0.083333, 0.291667 ] |
Frequently asked questions
What does a zero determinant mean?
The rows are linearly dependent: the matrix squashes space flat, so the system Ax = b has no unique solution.
Sources and further reading
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