Let | A |=6 , where A is a 3 3 matrix. If | adj (3 adj ( ~A ² adj (2 ~A ) ) ) |=2^ m 3^ n , m , n N , then m + n is equal to _ _ _ _ .

Step-by-step solution

For 3 3 matrix: | adj (M)| = |M|^2 , |kM| = k^3|M| . |2A| = 8 6 = 48 , | adj (2A)| = 48^2 = 2^8 3^2 . |A^2 adj (2A)| = |A|^2 | adj (2A)| = 2^2 3^2 2^8 3^2 = 2¹⁰ 3^4 . Let B = A^2 adj (2A) . | adj (B)| = |B|^2 = 2²⁰ 3^8 . |3 , adj (B)| = 3^3 2²⁰ 3^8 = 2²⁰ 3¹¹ . | adj (3 , adj (B))| = (2²⁰ 3¹¹)^2 = 2⁴⁰ 3²² . m = 40, n = 22 m + n = 62 .