Question
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 \_\_\_\_.
JEE Main
Mathematics
Matrices
2026
JEE Main 2026 (23 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
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 = 2^ 10 3^4. Let B = A^2 adj (2A). | adj (B)| = |B|^2 = 2^ 20 3^8. |3\, adj (B)| = 3^3 2^ 20 3^8 = 2^ 20 3^ 11 . | adj (3\, adj (B))| = (2^ 20 3^ 11 )^2 = 2^ 40 3^ 22 . m = 40, n = 22 m + n = 62.
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