If ij A [a_ij] is a matrix of order 2 × 2, such that |A|= -15 and C_ij represents the cofactor of , a_ij a then find a₂₁ c₂₁ + a₂₂ c₂₂.

Let A be the 2×2 matrix

And |A| = -15

⇒ a₁₁a₂₂ – a₁₂a₂₁ = -15 …(i)

Let M_ij represent minor of a_ij and C_ij is cofactor of a_ij.

M_ij is the determinant of matrix leaving the i^th row and j^th column

The cofactor is given by

⇒ C_ij = (-1) ^i+jM_ij

We have to find a₂₁C₂₁ + a₂₂C₂₂

C₂₁ = (-1)²⁺¹M₂₁ = -a₁₂ …since M₂₁ = a₁₂

C₂₂ = (-1)²⁺²M₂₂ = a₁₁ …since M₂₂ = a₁₁

⇒ a₂₁C₂₁ + a₂₂C₂₂ = a₂₁(-a₁₂) + a₂₂a₁₁

⇒ a₂₁C₂₁ + a₂₂C₂₂ = a₂₂a₁₁ – a₂₁a₁₂

Using (i)

⇒ a₂₁C₂₁ + a₂₂C₂₂ = -15