Let A be a n × n matrix such that A = 2 . If the determinant of the matrix Adj 2 . Adj 2 A - 1 is 2 84 , then n is equal to _____ .

Step-by-step solution

Given, A = 2 Now simplifying, Adj 2 . Adj 2   A - 1 = 2 · Adj 2   A - 1 n - 1 = 2 n Adj 2   A - 1 n - 1 = 2 n 2 A - 1 n - 1 n - 1 = 2 n n - 1 2 n A - 1 n - 1 n - 1 ∴       A - 1 = 1 A = 1 2 = 2 n n - 1 2 n - 1 n - 1 n - 1 = 2 n n - 1 + n - 1 3 = 2 84 Now comparing both side we get, n n - 1 + n - 1 3 = 84 ⇒ n - 1 n + n 2 - 2 n + 1 = 84 ⇒ n - 1 n 2 - n + 1 = 4 × 21 Now if n - 1 = 4 ⇒ n = 5 now checking n 2 - 3 n + 1 = 25 - 5 + 1 = 21 Hence, n = 5