Enter 3×3 Matrix Values
Use the grid below to calculate the inverse of any invertible 3×3 matrix.
Example Data Table
This sample uses a matrix with determinant 1, so the inverse is easy to verify.
| Example | Input Matrix | Determinant | Inverse Matrix |
|---|---|---|---|
| Worked Example | [[1, 2, 3], [0, 1, 4], [5, 6, 0]] | 1 | [[-24, 18, 5], [20, -15, -4], [-5, 4, 1]] |
Formula Used
1) Determinant of a 3×3 Matrix
For matrix A = [[a, b, c], [d, e, f], [g, h, i]],
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
2) Cofactor Matrix
Each cofactor is Cij = (-1)i+j × Mij, where Mij is the determinant of the 2×2 minor.
3) Adjugate Matrix
adj(A) is the transpose of the cofactor matrix.
4) Inverse Matrix
A-1 = adj(A) / det(A), provided det(A) ≠ 0.
How to Use This Calculator
- Enter all nine values of your 3×3 matrix.
- Choose how many decimal places you want in the output.
- Click Invert Matrix to compute the determinant and inverse.
- Review the determinant, cofactor matrix, adjugate matrix, and inverse matrix.
- Check the identity verification table to confirm the result.
- Use the CSV or PDF buttons to download your computed output.
FAQs
1) When does a 3×3 matrix have an inverse?
A 3×3 matrix has an inverse only when its determinant is not zero. If the determinant equals zero, the matrix is singular and cannot be inverted.
2) Why is the determinant important here?
The determinant tells you whether inversion is possible. It also scales the adjugate matrix in the inverse formula. A zero determinant stops the calculation immediately.
3) What is the adjugate matrix?
The adjugate matrix is the transpose of the cofactor matrix. It is a required step in the standard formula for finding the inverse of a square matrix.
4) Can this calculator handle decimal values?
Yes. You can enter integers, negatives, or decimal values. The calculator computes the determinant, cofactors, adjugate, and inverse using floating-point arithmetic.
5) What does the identity check show?
It multiplies the original matrix by the computed inverse. A correct inverse gives a result very close to the 3×3 identity matrix, allowing for small rounding differences.
6) Why do some inverse values look rounded?
Inverse entries are often fractional or repeating decimals. The calculator rounds them to your selected precision so the table remains readable and easy to export.
7) What happens if I enter a singular matrix?
The page shows a clear error message explaining that the determinant is zero. In that case, no inverse, graph, or export file is produced.
8) What can I do with the download options?
You can export your matrix values, determinant, inverse, and verification data as a CSV file or create a PDF summary for reports, homework, or documentation.