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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Matrices

NDA 2017 · Mathematics · Matrices

If B is a non-singular matrix and A is a square matrix, then the...

If BB is a non-singular matrix and AA is a square matrix, then the value of det⁡⁡(B−1AB)isequalto\operatorname\det \left(B^{-1} A B\right) is equal to

Options

  1. Adet⁡⁡(B)\operatorname\det(\mathrm{B})
  2. Bdet⁡⁡(A)\operatorname\det(\mathrm{A})
  3. Cdet⁡⁡(B−1)\operatorname\det \left(\mathrm{B}^{-1}\right)
  4. Ddet⁡⁡(A−1)\operatorname\det \left(\mathrm{A}^{-1}\right)

Correct answer

B. det⁡⁡(A)\operatorname\det(\mathrm{A})

Step-by-step solution

∣B−1AB∣=∣B−1∣∣A∣∣B∣=1∣B∣∣A∣∣B∣=∣A∣\begin{aligned} & \left|\mathrm{B}^{-1} \mathrm{AB}\right|=\left|\mathrm{B}^{-1}\right||\mathrm{A}||\mathrm{B}| \\ & =\frac{1}{|\mathrm{ B}|}|\mathrm{A}||\mathrm{B}| \\ & =|\mathrm{A}| \end{aligned}\because \quad\left|\mathrm{B}^{-1}\right|=\frac{1}{|\mathrm{ B}|}$
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