Determinants

A determinant is a square arrangement

A determinant is a square arrangement of numbers enclosed between vertical bars. For a 3×3 determinant D = |a₁₁ a₁₂ a₁₃; a₂₁ a₂₂ a₂₃; a₃₁ a₃₂ a₃₃|, el

Two main methods

Two main methods: (1) Cofactor expansion along first row: D = a₁₁|a₂₂ a₂₃; a₃₂ a₃₃| - a₁₂|a₂₁ a₂₃; a₃₁ a₃₃| + a₁₃|a₂₁ a₂₂; a₃₁ a₃₂| (2) Diagonal metho

Seven key properties

Seven key properties: (1) Value unchanged when rows↔columns (2) Sign changes when two rows/columns are interchanged (3) Zero value when two rows/colum

For system a₁x + b₁y +

For system a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃, solutions are x = Dₓ/D, y = Dᵧ/D, z = Dᵧ/D where D is coefficient determi

For triangle with vertices A(x₁

For triangle with vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃), Area = ½|x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1|. Always take absolute value as area is positive. Example: