Determinant Calculator

Find the determinant of a square matrix and see the working. Cofactor expansion for small matrices, row reduction for larger ones. Exact fractions, no rounding. Dimensions up to 6×6.

How to Find a Determinant

Enter the entries of a square matrix — whole numbers, decimals, and fractions like 1/2 are all accepted.

1

Set the size

Pick a size from 1×1 to 6×6. Determinants are only defined for square matrices, so rows and columns always match.

2

Fill in the entries

Type a value into each cell. The result updates as you type, and every value is kept as an exact fraction rather than a rounded decimal.

3

Read the result and steps

The determinant appears first. Matrices up to 3×3 show the cofactor expansion term by term; 4×4 and larger show the row-reduction elimination steps.

Frequently Asked Questions

What is the determinant of a matrix?
The determinant is a single number computed from a square matrix that describes how the matrix scales area or volume, and whether it flips orientation. A determinant of zero means the matrix is singular — it collapses space into a lower dimension and has no inverse. Determinants are only defined for square matrices.
How do you calculate a determinant by hand?
For a 2x2 or 3x3 matrix, cofactor expansion along the first row is common: multiply each entry by its signed minor determinant and add the results. For larger matrices, row reduction to an upper-triangular form is faster — the determinant is then the product of the diagonal entries, adjusted for any row swaps. This calculator uses cofactor expansion for matrices up to 3x3 and elimination for 4x4 and larger, showing every step.
What does a determinant of zero mean?
A determinant of zero means the matrix is singular: its rows (or columns) are linearly dependent, it has no inverse, and the linear system it represents either has no solution or infinitely many. Geometrically, a singular matrix squashes space into a lower dimension, so areas or volumes it maps become zero.
Why does this calculator use fractions instead of decimals?
Matrix entries are stored and combined as exact fractions using integer arithmetic, so results like 1/3 stay exact instead of rounding to 0.333... Determinant calculations involve many multiplications and additions, and rounding errors compound quickly in larger matrices, so exact fractions keep every step and the final result mathematically precise.