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JEE MAIN Mathematics QUESTION #962
Question 1
For a \(3 \times 3\) matrix \(M\), let trace\((M)\) denote the sum of all diagonal elements of \(M\). Let \(A\) be a \(3 \times 3\) matrix such that \(|A| = \dfrac{1}{2}\) and trace\((A) = 3\). If \(B = \text{adj}(\text{adj}(2A))\), then the value of \(|B| + \text{trace}(B)\) equals:
  • 56
  • 132
  • 174
  • 280✔️
Correct Answer Explanation
For \(n=3\): \(\text{adj}(\text{adj}(M)) = |M|^{n-2} M\). So \(\text{adj}(\text{adj}(2A)) = |2A|^1 \cdot 2A = 8|A| \cdot 2A = 8A\). Thus \(|B| = |8A| = 8^3|A| = 512 \cdot \frac{1}{2} = 256\) and \(\text{trace}(B) = \text{trace}(8A) = 8\cdot3 = 24\). So \(|B| + \text{trace}(B) = 256 + 24 = 280\).