Let A = bmatrix 3 & 1 & 2 1 & 2 & 0 1 & 1 & 4 bmatrix and pC₁₁ + 4C₂₁ - 5C₃₂ = -2 , where C_ ij denotes the cofactor of an element a_ ij of matrix A , then the value of p is :

Step-by-step solution

Given matrix A = bmatrix 3 & 1 & 2 1 & 2 & 0 1 & 1 & 4 bmatrix The cofactor C_ ij is given by (-1)^ i+j M_ ij , where M_ ij is the minor of the element a_ ij . Calculating the required cofactors: C₁₁ = (-1)¹⁺¹ vmatrix 2 & 0 1 & 4 vmatrix = 1 (8 - 0) = 8 C₂₁ = (-1)²⁺¹ vmatrix 1 & 2 1 & 4 vmatrix = -1 (4 - 2) = -2 C₃₂ = (-1)³⁺² vmatrix 3 & 2 1 & 0 vmatrix = -1 (0 - 2) = 2 Substituting these values into the given equation pC₁₁ + 4C₂₁ - 5C₃₂ = -2 : p(8) + 4(-2) - 5(2) = -2 8p - 8 - 10 = -2 8p - 18 = -2 8p = 16 p = 2 An