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NDA 2023 Maths · Mathematics · Determinants · hard

If A is a 3×3 matrix with det A=-2, then det (adj(A^2) equals

Single correct option

If AA is a 3×33\times3 matrix with det⁡A=−2\det A=-2, then det⁡(adj⁡(A2))\det\big(\operatorname{adj}(A^2)\big) equals
  1. A. −16-16
  2. B. 44
  3. C. 1616
  4. D. −4-4
Show answer and solution

Answer: (C) 1616

For an n×nn\times n matrix BB, det⁡(adj⁡B)=(det⁡B)n−1\det(\operatorname{adj}B)=(\det B)^{n-1}. Here B=A2B=A^2 has det⁡B=(−2)2=4\det B=(-2)^2=4, so det⁡(adj⁡B)=42=16\det(\operatorname{adj}B)=4^{2}=16. Option 44 forgets the exponent n−1=2n-1=2, and the negative values result from carrying the sign of det⁡A\det A through an even power.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake