Let A_k = bmatrix 2^k & 1 & 1 2^k & 2 & 1 2^k & 1 & 2 bmatrix . If the sum S = _ k=1 ¹⁰ ( adj (2 A_k)) is expressed in the form 2^a 3 (2^b - 1) , where a and b are integers, then the value of a + b is

Step-by-step solution

First, we find the determinant of A_k . Taking 2^k common from the first column: (A_k) = 2^k vmatrix 1 & 1 & 1 1 & 2 & 1 1 & 1 & 2 vmatrix Applying the row operations R₂ R₂ - R₁ and R₃ R₃ - R₁ : (A_k) = 2^k vmatrix 1 & 1 & 1 0 & 1 & 0 0 & 0 & 1 vmatrix = 2^k (1) = 2^k Using the property of determinants for a 3 3 matrix, (cA) = c^3 (A) : (2 A_k) = 2^3 (A_k) = 8 2^k = 2^ k+3 The determinant of the adjoint of a 3 3 matrix is the square of its determinant: ( adj (2 A_k)) = ( (2 A_k))^2 = (2^ k+3 )^2 = 2^ 2k+6 = 64 4^k