Determinants

Definition of a Determinant

• A determinant is a square array of numbers or expressions arranged in rows and columns.
• The number of rows equals the number of columns.
• The order of a determinant is the number of rows (or columns).

Representation of a Determinant

• Denoted by |A| or Δ
• For a 3 × 3 determinant: | a₁₁ a₁₂ a₁₃ ; a₂₁ a₂₂ a₂₃ ; a₃₁ a₃₂ a₃₃ |
• aᵢⱼ denotes element of i-th row and j-th column
• Elements where i = j are called diagonal elements

Special Determinants

• Triangular Determinant: All elements above or below the principal diagonal are zero
• Diagonal Determinant: All non-diagonal elements are zero
• Value of a diagonal determinant = product of diagonal elements

Value of a Determinant

• The numerical value obtained after expansion is called the value of the determinant
• A determinant of order 3 can be expanded along any row or column

Minor of an Element

• Minor of element aᵢⱼ is obtained by deleting the i-th row and j-th column
• Minor is denoted by Mᵢⱼ

Cofactor of an Element

• Cofactor of aᵢⱼ is denoted by Cᵢⱼ
• Cᵢⱼ = (−1)^i+j Mᵢⱼ

Expansion of a Determinant

• Expansion along first row: Δ = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃
• Expansion along any row or column gives the same value
• Sum of products of elements of a row with cofactors of another row is zero

Important Properties of Determinants

• Interchanging rows and columns does not change the value
• Interchanging two rows (or columns) changes the sign of the determinant
• If two rows or columns are identical, determinant value is 0
• If each element of a row or column is multiplied by k, determinant is multiplied by k
• If a row (or column) is expressed as sum of two rows, determinant can be split into sum of two determinants
• Adding a multiple of one row to another does not change the value
• If all elements of a row or column are zero, determinant value is zero

Factor Property

• If Δ = f(x) and f(a) = 0, then (x − a) is a factor of Δ

Multiplication of Determinants

• Multiplication is defined only when determinants are of the same order
• Resultant determinant is formed using row-by-column multiplication

Symmetric & Skew-Symmetric Determinants

• Symmetric: aᵢⱼ = aⱼᵢ
• Skew-symmetric: aᵢⱼ = −aⱼᵢ
• Diagonal elements of a skew-symmetric determinant are zero
• Skew-symmetric determinant of odd order is always zero
• Skew-symmetric determinant of even order is a perfect square

Applications of Determinants — Cramer’s Rule

• For system of linear equations in three variables:
• Δ = | a₁ b₁ c₁ ; a₂ b₂ c₂ ; a₃ b₃ c₃ |
• x = Δ₁ / Δ, y = Δ₂ / Δ, z = Δ₃ / Δ
• If Δ ≠ 0 → unique solution
• If Δ = 0 and at least one Δᵢ ≠ 0 → no solution
• If Δ = Δ₁ = Δ₂ = Δ₃ = 0 → infinitely many solutions

Special Cases of Solutions

• If Δ ≠ 0 and all constants are zero → trivial solution
• If Δ = 0 and constants are zero → non-trivial solutions exist

Differentiation of a Determinant

• Differentiate one row at a time, keeping others constant
• Sum of all such determinants gives derivative

Integration of a Determinant

• Integral is taken row-wise
• Constants are factored outside the determinant

Use of Summation in Determinants

• Summation terms can be taken column-wise or row-wise
• Useful in problems involving series and determinants

JEE Main Focus Tips

• Use properties to simplify before expansion
• Avoid direct expansion when rows/columns are similar
• Check for zero determinant using row proportionality
• Cramer’s rule questions are frequent and scoring