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Let A be a 3 × 3 matrix and det A = 2 . If n = det a d j a d j . ... a d j A ⏟ 2024 − times , then the remainder when n is divided by 9 is equal to __________.

Correct answer

0

Step-by-step solution

Given, A = 2 And n = det a d j a d j a d j . .. A ⏟ 2024 times Now, using the formula a d j a d j a d j . .. A ⏟ r times = A n − 1 r we get, a d j a d j a d j . .. A ⏟ 2024 times = A n − 1 2024 ⇒ n = A 2 2024 ⇒ n = 2 2 2024 as A = 2 Now, solving 2 2024 = 4 1012 = 3 + 1 1012 = 3 k + 1 ⇒ 2 2 2024 = 2 3 k + 1 ⇒ 2 2 2024 = 2 · 2 3 k ⇒ 2 2 2024 = 2 · 9 - 1 k ⇒ 2 2 2024 = 9 m - 2 ⇒ 2 2 2024 = 9 t - 2 + 9 ⇒ 2 2 2024 = 9 m + 7 Hence, the remainder is 7

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