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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Determinants

NDA 2013 · Mathematics · Determinants

If D is a determinant of order 3 and D^ is the determinant obtained...

If DD is a determinant of order 3 and D′D^{\prime} is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct?

Options

  1. AD′=D2D^{\prime}=D^2
  2. BD′=D3\mathrm{D}^{\prime}=\mathrm{D}^3
  3. CD′=3D2\mathrm{D}^{\prime}=3 \mathrm{D}^2
  4. DD′=3D3\mathrm{D}^{\prime}=3 \mathrm{D}^3

Correct answer

A. D′=D2D^{\prime}=D^2

Step-by-step solution

D′= cofactor D∣D′∣=∣ cofactor D∣∣D′∣=∣D∣3−1∣D′∣=∣D∣2\begin{aligned} & \mathrm{D}^{\prime}=\text{ cofactor } \mathrm{D} \\ & \left|\mathrm{D}^{\prime}\right|=\mid \text{ cofactor } \mathrm{D} \mid \\ & \left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{3-1} \\ & \left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^2\end{aligned} So D′=D2\mathrm{D}^{\prime}=\mathrm{D}^2
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