---
title: "Matrix Determinant Calculator — 2×2 to 5×5 with Step-by-Step"
description: "Calculate the determinant of any square matrix from 2×2 to 5×5. Shows cofactor expansion for small matrices and LU decomposition for larger ones. Includes singularity check and geometric interpretation."
url: "https://worldcalculators.org/calculators/matrix-determinant/"
canonical: "https://worldcalculators.org/calculators/matrix-determinant/"
---

# Matrix Determinant Calculator

Calculate the determinant of any square matrix from 2×2 to 5×5. Shows cofactor expansion for small matrices and LU decomposition for larger ones. Includes singularity check and geometric interpretation.

## Quick answer

The Sarrus rule is a mnemonic for computing 3×3 determinants: write the matrix, repeat the first two columns to the right, then sum the three main diagonals (going down-right) and subtract the three anti-diagonals (going down-left). It is only valid for 3×3 and is popular in German, French, and Spanish high school curricula. It does NOT generalize to 4×4 or larger matrices — a common mistake is to attempt the "Sarrus method" on 4×4 matrices.

## Frequently Asked Questions

The Sarrus rule is a mnemonic for computing 3×3 determinants: write the matrix, repeat the first two columns to the right, then sum the three main diagonals (going down-right) and subtract the three anti-diagonals (going down-left). It is only valid for 3×3 and is popular in German, French, and Spanish high school curricula. It does NOT generalize to 4×4 or larger matrices — a common mistake is to attempt the "Sarrus method" on 4×4 matrices.
Cramer's rule solves a linear system Ax = b by computing xᵢ = det(Aᵢ) / det(A), where Aᵢ is A with column i replaced by b. It requires det(A) ≠ 0 (non-singular) and is computationally expensive for large systems (O(n!) operations), so it's mainly used for 2×2 and 3×3 systems in exams. Gaussian elimination is preferred computationally.

Source: https://worldcalculators.org/calculators/matrix-determinant/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.