Determinant Calculator

Understanding Determinants

The determinant is a scalar value that provides important information about a square matrix. It encodes properties like invertibility, volume scaling, and orientation.

Mathematical Definition

For an n×n matrix A, the determinant det(A) or |A| is a scalar that can be calculated recursively using cofactor expansion.

2×2 Matrix

For a 2×2 matrix:

3×3 Matrix

For a 3×3 matrix using the first row expansion:

where Cᵢⱼ are the cofactors.

Properties of Determinants

• Multiplicative: det(AB) = det(A)det(B)
• Transpose: det(Aᵀ) = det(A)
• Inverse: det(A⁻¹) = 1/det(A)
• Scalar multiple: det(kA) = kⁿdet(A) for n×n matrix
• Row operations: Swapping rows changes sign

Geometric Interpretation

• 2D: Area of parallelogram formed by column vectors
• 3D: Volume of parallelepiped formed by column vectors
• General: n-dimensional volume scaling factor
• Sign: Indicates orientation (positive = same, negative = reversed)

Applications

Linear Systems

• Cramer's rule for solving systems
• Determining if system has unique solution
• Matrix invertibility (det ≠ 0)

Calculus and Analysis

• Jacobian determinants in change of variables
• Volume integrals and transformations
• Differential equations and stability

Geometry

• Area and volume calculations
• Orientation preservation in transformations
• Cross products in vector algebra