3×3 Determinant Calculator
Enter the nine values of your matrix below. Decimals and negative numbers are supported.
For A = [[a, b, c], [d, e, f], [g, h, i]],
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
What is the determinant of a 3×3 matrix?
The determinant is a single number computed from the entries of a square matrix. For a 3×3 matrix, the determinant tells you important properties of the transformation represented by that matrix. In practical terms, it helps answer questions like:
- Is the matrix invertible?
- Does a system of three linear equations have a unique solution?
- How does the matrix scale 3D volume?
If the determinant is zero, the matrix is singular (not invertible). If it is non-zero, the matrix is invertible.
How this determinant calculator works
This calculator uses expansion along the first row: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg). After you enter values and click calculate, it computes each minor term and then combines them to produce the final determinant.
Benefits of using the tool
- Fast and accurate arithmetic for integer and decimal entries
- Step display so you can see how the result is built
- Great for homework checks, exam prep, and engineering workflows
Interpreting the result
When det(A) = 0
The rows (or columns) are linearly dependent. Geometrically, a 3D object transformed by this matrix is flattened into a lower-dimensional shape, so volume collapses to zero.
When det(A) > 0
The matrix preserves orientation and scales volume by a positive factor equal to the determinant.
When det(A) < 0
The matrix reverses orientation and scales volume by the absolute value of the determinant.
Worked example
Suppose A = [[1, 2, 3], [0, 4, 5], [1, 0, 6]]. Then:
- ei − fh = 4·6 − 5·0 = 24
- di − fg = 0·6 − 5·1 = -5
- dh − eg = 0·0 − 4·1 = -4
So det(A) = 1(24) − 2(-5) + 3(-4) = 24 + 10 − 12 = 22. Because 22 is non-zero, the matrix is invertible.
Common mistakes to avoid
- Mixing up the minus sign in the middle term: − b(di − fg)
- Entering row/column values in the wrong position
- Forgetting that decimal rounding can affect hand calculations
- Assuming a small determinant means zero—it does not unless it is exactly zero
Where 3×3 determinants are used
Linear algebra and systems
Determinants appear in Cramer’s Rule and matrix invertibility tests for solving linear systems.
Computer graphics and robotics
They are used to detect orientation flips and evaluate transformation properties in 3D coordinate frames.
Physics and engineering
Determinants show up in Jacobians, change-of-variable formulas, and many modeling computations.
Quick FAQ
Can I enter fractions?
Yes—enter them as decimals (for example, 1/2 as 0.5).
Does order matter?
Absolutely. Changing entry positions changes the determinant.
Can this calculator solve eigenvalues?
No, this page is focused specifically on the determinant of a 3×3 matrix.