Let A be a 3 3 matrix with real entries such that | adj ( A adj (2A) ) | = 2¹² 5^6 . Then the value of | 2 A^2 | is equal to

Step-by-step solution

Let M = A adj (2A) . We know that for a 3 3 matrix X , |kX| = k^3 |X| and | adj (X)| = |X|^2 . First, find the determinant of M : |M| = |A| | adj (2A)| |2A| = 2^3 |A| = 8|A| | adj (2A)| = |2A|^2 = (8|A|)^2 = 64|A|^2 = 2^6 |A|^2 So, |M| = |A| (2^6 |A|^2) = 2^6 |A|^3 . The given expression is | adj (M)| : | adj (M)| = |M|^2 = (2^6 |A|^3)^2 = 2¹² |A|^6 . Equating this to the given value: 2¹² |A|^6 = 2¹² 5^6 |A|^6 = 5^6 |A|^2 = 25 (since A has real entries, |A| is a real number). We need to find the value of |2A^2| : |