Let A be a non-singular square matrix of order 3 such that adj (2A) = 8 A^T . If det ( adj (2A) adj (4 A^T)) = 2^m , then the value of m is equal to _______

Step-by-step solution

For a square matrix of order n , adj (kA) = k^ n-1 adj (A) . Here n=3 , so adj (2A) = 2^2 adj (A) = 4 adj (A) . Given adj (2A) = 8 A^T , we have: 4 adj (A) = 8 A^T adj (A) = 2 A^T Taking the determinant on both sides: det ( adj (A)) = det (2 A^T) ( det (A))^2 = 2^3 det (A^T) Since det (A^T) = det (A) and A is non-singular ( det (A) 0 ), we get: det (A) = 8 = 2^3 Now, evaluate the given expression: det ( adj (2A) adj (4 A^T)) = det ( adj (2A)) det ( adj (4 A^T)) First term: det ( adj (2A)) = ( det (2A))^2 = (2^3 det