Let A = 0 1 2 a 0 3 1 c 0 , where a , c ∈ R . If A 3 = A and the positive value of a belongs to the interval ( n - 1 , n ] , where n ∈ ℕ , then n is equal to ____.

Step-by-step solution

Given, A = 0 1 2 a 0 3 1 c 0 ⇒ A 2 = 0 1 2 a 0 3 1 c 0 0 1 2 a 0 3 1 c 0 ⇒ A 2 = a + 2 2 c 3 3 a + 3 c 2 a c a 1 2 + 3 c Now finding, A 3 = a + 2 2 c 3 3 a + 3 c 2 a c a 1 2 + 3 c 0 1 2 a 0 3 1 c 0 ⇒ A 3 = 2 c a + 3 α + 2 + 3 c 2 a + 4 + 6 c a 2 + 3 c a + 2 a 3 + 2 a c 6 + 3 a + 9 c a + 2 + 3 c c a + 2 c + 3 c 2 2 c a + 3 Now equating A 3 = A and comparing both side we get, 2 c a + 3 = 0 ⇒ c = - 3 2 a And a + 2 + 3 c = 1 ⇒ a + 2 + 3 - 3 2 a = 1 ⇒ a + 1 - 9 2 a = 0 ⇒ 2