Determinant Calculator

About Determinant Calculator

The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. It is denoted as det(A), det A, or |A| for a matrix A.

How Determinant is Calculated

The method for calculating the determinant depends on the size of the matrix:

• 2×2 Matrix: For matrix [[a, b], [c, d]], the determinant is ad - bc
• 3×3 Matrix: Using the rule of Sarrus or Laplace expansion
• 4×4 and larger: Typically calculated using Laplace expansion (cofactor expansion) or row reduction

Properties of Determinants

Determinants have several important mathematical properties:

• The determinant of the identity matrix is 1
• If two rows or columns are identical, the determinant is 0
• Swapping two rows or columns changes the sign of the determinant
• The determinant of a product of matrices equals the product of their determinants
• A matrix is invertible if and only if its determinant is non-zero

Applications of Determinants

Determinants are used in various areas of mathematics and applications:

• Solving systems of linear equations (Cramer's Rule)
• Finding the inverse of a matrix
• Determining if a matrix is invertible (non-singular)
• Calculating eigenvalues and eigenvectors
• Computing volumes and areas in linear transformations
• In physics for calculating cross products and moments

Special Cases

Some matrices have special determinant properties:

• Diagonal Matrix: Determinant is the product of diagonal elements
• Triangular Matrix: Same as diagonal matrix
• Orthogonal Matrix: Determinant is ±1
• Singular Matrix: Determinant is 0 (matrix is not invertible)