Determinants and their Applications

For a 2×2 matrix A =

For a 2×2 matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], the determinant is |A| = a₁₁a₂₂ - a₂₁a₁₂. This is calculated by multiplying diagonal elements and subtr

For a 3×3 matrix

For a 3×3 matrix, the determinant can be calculated by expanding along any row or column using cofactors. The general formula involves six terms: thre

Minor Mᵢⱼ of element aᵢⱼ

Minor Mᵢⱼ of element aᵢⱼ is the determinant obtained by deleting the i-th row and j-th column. Cofactor Cᵢⱼ = (-1)^(i+j) × Mᵢⱼ. The sign pattern alter

Alternative method for 3×3 determinants

Alternative method for 3×3 determinants: Write the first two columns again after the original matrix. Draw diagonal arrows downward (positive terms) a

Key properties include

Key properties include: (1) Interchanging rows/columns changes sign, (2) Identical rows/columns make determinant zero, (3) Multiplying a row by k mult