3×3 Determinant Calculator
Calculate the determinant of a 3×3 matrix step by step. Free 3×3 determinant calculator with cofactor expansion, exact fractions, and a worked example that answers -306.
Find the determinant of any 3×3 matrix with a full worked solution. The grid below is preset to 3×3 and loaded with an example whose determinant is -306.
How to find the determinant of a 3×3 matrix
- 1
Enter the entries
Fill in all 9 numbers of your 3×3 matrix. The size is already set for you.
- 2
Expand the cofactors
The calculator multiplies each first-row entry by its minor and alternates the signs.
- 3
Read the determinant
The result appears at the top, together with whether the 3×3 matrix is invertible.
About 3×3 determinants
A 3×3 determinant is usually done by cofactor expansion along a row, or with the Rule of Sarrus.
The example above works out to -306. Because it is non-zero, that matrix is invertible and its rows are linearly independent. Change any entry and the whole solution updates as you type.
Other matrix sizes
Frequently asked questions
How do you find the determinant of a 3×3 matrix? ▾
Expand along the first row: each of the three entries is multiplied by the 2x2 determinant left after deleting its row and column, and the terms alternate plus, minus, plus. The Rule of Sarrus gives the same result.
How many terms does a 3×3 cofactor expansion have? ▾
A full cofactor expansion of an 3x3 matrix has 6 products. That is why this calculator uses cofactor expansion.
What does it mean if a 3×3 determinant is zero? ▾
A zero determinant means the 3 rows are linearly dependent — one is a combination of the others. The matrix is singular, has no inverse, and the associated system of 3 equations does not have a unique solution.
Can this 3×3 determinant calculator handle fractions? ▾
Yes. Type entries like 3/4 or -1/2 and the answer comes back as an exact fraction rather than a rounded decimal.
About this Calculator
Calculate the determinant of a 3×3 matrix step by step. Free 3×3 determinant calculator with cofactor expansion, exact fractions, and a worked example that answers -306.