Let A be a matrix of order 3 3 such that A + I = bmatrix +1 & 2 & 1 1 & +1 & 0 0 & 1 & 2 bmatrix , where I is the identity matrix of order 3 3 and R . If (3 adj (2 adj (A))) = 2^6 3^7 , then the sum of the fourth powers of all possible real values of is equal to :

Step-by-step solution

A = bmatrix +1 & 2 & 1 1 & +1 & 0 0 & 1 & 2 bmatrix - bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 bmatrix = bmatrix & 2 & 1 1 & & 0 0 & 1 & 1 bmatrix (A) = ( - 0) - 2(1 - 0) + 1(1 - 0) = ^2 - 1 Using the properties of determinants and adjoints for a 3 3 matrix: (k M) = k^3 (M) ( adj (M)) = ( (M))^2 Let us simplify the given expression: (3 adj (2 adj (A))) = 3^3 ( adj (2 adj (A))) = 27 ( (2 adj (A)))^2 = 27 (2^3 ( adj (A)))^2 = 27 (8 ( (A))^2)^2 = 27 64 ( (A))^4 = 3^3 2^6 ( ^2 - 1)^4 We are given that this determinant is