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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Matrices

NDA 2015 · Mathematics · Matrices

If A is an invertible matrix of order n and k is any positive real...

If AA is an invertible matrix of order n and k is any positive real number, then the value of [det⁡⁡(kA)]−1det⁡⁡A[\operatorname\det(k A)]^{-1} \operatorname\det A is

Options

  1. A\mathrmk−n\mathrmk^{-\mathrm{n}}
  2. B\mathrmk−1\mathrmk^{-1}
  3. C\mathrmkn\mathrmk^{\mathrm{n}}
  4. Dnk

Correct answer

A. \mathrmk−n\mathrmk^{-\mathrm{n}}

Step-by-step solution

If A is matrix that is invertible then det- ( kA ) will be \mathrmkn.det⁡(A)\mathrmk^{\mathrm{n}} . \operatorname{det}(\mathrm{A}), where n is the order. ∴[det⁡(KA)]−1det⁡(A)=[(K)n×det⁡(A)]−1⋅det⁡(A)=K−n×1det⁡(A)×det⁡(A)=K−n\begin{aligned} & \therefore[\operatorname{det}(\mathrm{KA})]^{-1} \operatorname{det}(\mathrm{ A}) \\ & =\left[(\mathrm{K})^{\mathrm{n}} \times \operatorname{det}(\mathrm{A})\right]^{-1} \cdot \operatorname{det}(\mathrm{ A}) \\ & =\mathrm{K}^{-\mathrm{n}} \times \frac{1}{\operatorname{det}(\mathrm{ A})} \times \operatorname{det}(\mathrm{A}) \\ & =\mathrm{K}^{-\mathrm{n}} \end{aligned}
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