Determinant Calculator

Matrix A — size × size

How to Find a Determinant

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

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.

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.

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 has either no solution or infinitely many. Geometrically, a singular matrix squashes space into a lower dimension, so the area or volume it maps becomes zero. The matrix [[1, 2], [2, 4]] has determinant 1×4 − 2×2 = 0, because the second row is just twice the first. Any non-zero determinant, however small, means the matrix is invertible.

Why does this calculator use fractions instead of decimals?

Matrix entries are stored and combined as exact fractions using integer arithmetic, so a value like 1/3 stays exact instead of rounding to 0.3333. Determinant calculations involve many multiplications and subtractions, and floating-point rounding errors compound quickly in larger matrices. A matrix that is genuinely singular can then report a determinant of 0.0000000000000001 rather than 0, which answers a different question from the one you asked. Exact fractions keep every intermediate step and the final determinant mathematically precise.

How to Find a Determinant

Set the size

Fill in the entries

Read the result and steps

Frequently Asked Questions

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