Determinants

A determinant is a special number

A determinant is a special number (real or complex) associated with every square matrix. For a square matrix A, the determinant is denoted as |A|, det

For a 1×1 matrix A =

For a 1×1 matrix A = [a], the determinant is simply the element itself: |A| = a. This is the simplest case of determinant calculation.

For a 2×2 matrix A =

For a 2×2 matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], the determinant is calculated as: |A| = a₁₁a₂₂ - a₁₂a₂₁. This follows the cross-multiplication pattern:

For a 3×3 matrix

For a 3×3 matrix, determinants are calculated by expansion along any row or column. Expanding along the first row: |A| = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂

The minor Mᵢⱼ of element aᵢⱼ

The minor Mᵢⱼ of element aᵢⱼ is the determinant obtained by deleting the i-th row and j-th column. The cofactor Aᵢⱼ = (-1)^(i+j) × Mᵢⱼ. Cofactors incl