5×5 Determinant Calculator
Calculate the determinant of a 5×5 matrix step by step. Free 5×5 determinant calculator with cofactor expansion, exact fractions, and a worked example that answers 31.
Find the determinant of any 5×5 matrix with a full worked solution. The grid below is preset to 5×5 and loaded with an example whose determinant is 31.
How to find the determinant of a 5×5 matrix
- 1
Enter the entries
Fill in all 25 numbers of your 5×5 matrix. The size is already set for you.
- 2
Reduce to triangular form
Row operations clear each column below the pivot until the matrix is upper triangular.
- 3
Read the determinant
The result appears at the top, together with whether the 5×5 matrix is invertible.
About 5×5 determinants
A 5×5 cofactor expansion has 120 terms, so the calculator switches to row reduction automatically.
The example above works out to 31. 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 5×5 matrix? ▾
Row-reduce the matrix to upper triangular form, tracking each row swap as a sign change, then multiply the diagonal entries. Cofactor expansion would need 120 products, so reduction is much faster.
How many terms does a 5×5 cofactor expansion have? ▾
A full cofactor expansion of an 5x5 matrix has 120 products. That is why this calculator uses row reduction to triangular form.
What does it mean if a 5×5 determinant is zero? ▾
A zero determinant means the 5 rows are linearly dependent — one is a combination of the others. The matrix is singular, has no inverse, and the associated system of 5 equations does not have a unique solution.
Can this 5×5 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 5×5 matrix step by step. Free 5×5 determinant calculator with cofactor expansion, exact fractions, and a worked example that answers 31.