Doubtify – JEE Ki Taiyaari, Simple Bhasha Mein

  ❓ Question Let A A be a 3 × 3 3 \times 3 3 × 3 matrix such that ∣ adj ⁡ ( adj ⁡ ( adj ⁡ A ) ) ∣ = 81. If S = { n ∈ Z : ∣ adj ⁡ ( adj ⁡ A ) ∣ ( n − 1 ) 2 2 = ∣ A ∣   3 n 2 − 5 n − 4 } , then ∑ n ∈ S ∣ A ∣   n 2 + n is equal to ? 🖼️ Question Image ✍️ Short Solution We use two standard facts for an invertible n × n n \times n  matrix A A : det ⁡ ( adj ⁡ A ) = ( det ⁡ A ) n − 1 \det(\operatorname{adj} A) = (\det A)^{n-1} For 3×3, n = 3 ⇒ det ⁡ ( adj ⁡ A ) = ∣ A ∣ 2 n = 3 \Rightarrow \det(\operatorname{adj}A) = |A|^2 From ∣ adj ⁡ ( adj ⁡ ( adj ⁡ A ) ) ∣ = 81 |\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))| = 81 , we express everything in terms of ∣ A ∣ |A| , then solve the given equation for integers n n . Finally we compute ∑ n ∈ S ∣ A ∣ n 2 + n \sum_{n\in S} |A|^{n^2+n} 🔹 Step 1 — Express all determinants in terms of ∣ A ∣ |A| Let ∣ A ∣ = D . For a 3×3 matrix: First adjoint: B 1 = adj ⁡ A , ∣ B 1 ∣ = ∣ A ∣ 2 = D 2 . Second adjoint...