What does it mean if the determinant is zero?

A zero determinant indicates a singular (non-invertible) matrix that collapses space onto a lower dimension, meaning its system of equations has no unique solution.

What is Cramer's rule?

Cramer's rule is an explicit formula for solving a system of linear equations using determinants of the coefficient and replaced-column matrices.

Can a determinant be negative?

Yes, a negative determinant indicates that the linear transformation reverses the spatial orientation (like a reflection).

Mathematics Tools• Linear Algebra

Matrix Determinant Calculator (2x2 & 3x3 Linear Algebra)

Determinants are foundational in linear algebra, physics, 3D graphics transformations, and system solving (Cramer's Rule). Compute determinants with full expansion steps.

100% private in-browser computation.

Determinant det(A)

det(A) = 22

1(4*6 - 5*0) - 2(0*6 - 5*1) + 3(0*0 - 4*1) = 22

Matrix is Invertible (det ≠ 0)

How to Use Matrix Determinant Calculator

Select Matrix Size

Choose 2x2 or 3x3 square matrix dimension.

Enter Matrix Elements

Type numerical values into the grid cells.

Inspect Determinant

Review det(A), step-by-step formula expansion, and invertibility status.

About Matrix Determinant Calculator

For a 2x2 matrix [[a, b], [c, d]], det = ad - bc. For a 3x3 matrix, evaluates Laplace cofactor expansion along the top row: a(ei - fh) - b(di - fg) + c(dh - eg).

Supported Formats

Real NumbersDecimalsIntegers

Operational Limits

•Calculates determinants for 2x2 and 3x3 square matrices.

Key Features

Instant 2x2 (ad - bc) and 3x3 cofactor expansion computation
Full step-by-step breakdown showing minor matrices and signed cofactors
Invertibility check (singular vs non-singular matrix, det != 0)
Supports positive, negative, and decimal entries
Clean mathematical notation output

Frequently Asked Questions

In 2D, the absolute determinant represents the scaling factor of area; in 3D, it represents the scaling factor of volume transformed by the matrix.

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