Let det(adj(adjA)) = 14^4 where A = <table><tr><td>x</td><td>2</td><td>-1</td></tr><tr><td>-1</td><td>1</td><td>2</td></tr><tr><td>2</td><td>-1</td><td>1</td></tr></table>, x ≠ -25/3, then
Step-by-Step Solution
Key Concept: Use the property det(adj(adjA)) = (det A)^((n-1)^2) where n is the order of the matrix. Here n=3, so det(adj(adjA)) = (det A)^4. Given det(adj(adjA)) = 14^4, we have det A = 14 or -14.
Given A = [[x, 2, -1], [-1, 1, 2], [2, -1, 1]]. det(A) = x(1+2) - 2(-1-4) - 1(1-2) = 3x + 10 + 1 = 3x + 11. Since det(adj(adjA)) = (det A)^4 = 14^4, det A = 14 or -14. Case 1: 3x + 11 = 14 => 3x = 3 => x = 1. Then det(2A) = 2^3 * det(A) = 8 * 14 = 112. Case 2: 3x + 11 = -14 => 3x = -25 => x = -25/3 (rejected). Thus x=1 and det(2A)=112.
Correct Answer: 1, 2