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AlgebraFree

Determinant Calculator

Compute the determinant of a 2x2, 3x3, or 4x4 matrix from a single grid.

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Result

Enter values above and click Calculate to see your result.

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Formula

Cofactor expansion (recursive)

Select a size, enter entries, then evaluate the determinant with invertibility feedback.

Worked Example

Choose 3x3, enter nine entries, then read the determinant result.

What Determinants Tell You About a Matrix

The determinant is a single number that encodes fundamental properties of a square matrix. It tells you whether the matrix is invertible, how it scales volumes, and whether it preserves or reverses orientation.
  • Nonzero determinant: the matrix is invertible, the system Ax = b has a unique solution, and the rows (or columns) are linearly independent
  • Zero determinant: the matrix is singular, no inverse exists, and the system either has no solution or infinitely many
  • Geometric meaning: the absolute value of the determinant equals the scaling factor for area (2D) or volume (3D) under the linear transformation
  • Negative determinant: the transformation reverses orientation (like a mirror reflection)

This calculator computes determinants using cofactor expansion, the standard method taught in linear algebra courses.

You can also calculate changes using our Matrix Determinant Calculator, Determinant Calculator with Steps, 3x3 Determinant Calculator or 2x2 Determinant Calculator.

Frequently Asked Questions

How is this different from the other determinant calculators?

This is a single-page tool with a short URL optimized for the search term "determinant calculator." It uses the same cofactor expansion math as the dedicated 2x2, 3x3, and 4x4 calculators.

What if my matrix is singular?

The determinant equals zero, and the calculator reports that the matrix is not invertible. This means the rows are linearly dependent and the system has no unique solution.

Does a negative determinant mean an error?

No. Negative determinants are normal and mean the matrix reverses orientation (like reflecting across an axis). Only the sign and whether the value is zero or nonzero matter for most applications.

Which size should I select?

Choose 2x2 if you have 4 entries, 3x3 for 9 entries, or 4x4 for 16 entries. Extra fields are ignored for smaller matrix sizes.

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Matrix Determinant CalculatorDeterminant Calculator with Steps3x3 Determinant Calculator2x2 Determinant Calculator

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