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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Determinants

NDA 2023 · Mathematics · Determinants

Let A be a skew-symmetric matrix of order 3. What is the value of...

Let A be a skew-symmetric matrix of order 3. What is the value of det⁡⁡(4A4)−det⁡⁡(3A3)+det⁡⁡(2A2)−det⁡⁡(A)+det⁡⁡(−I)\operatorname\det\left(4 A^4\right)-\operatorname\det\left(3 A^3\right)+\operatorname\det\left(2 A^2\right)-\operatorname\det(A)+\operatorname\det(-I) Where I is the identity matrix of order 3 ?

Options

  1. A-1
  2. B0
  3. C1
  4. D2

Correct answer

A. -1

Step-by-step solution

Given AA is skew symmetric matrix of order 3 (odd) So ∣A∣=0|\mathrm{A}|=0 Now, ∣4A4∣−∣3A3∣+∣2A2∣−∣A∣+∣−I∣\left|4 A^4\right|-\left|3 A^3\right|+\left|2 A^2\right|-|A|+|-I|=43∣ A∣4−33∣ A∣3+22∣ A∣2−∣A∣+(−1)3∣I∣=0−0+0−0−1=−1\begin{aligned} & =4^3|\mathrm{~A}|^4-3^3|\mathrm{~A}|^3+2^2|\mathrm{~A}|^2-|\mathrm{A}|+(-1)^3|\mathrm{I}| \\ & =0-0+0-0-1=-1 \end{aligned}$$
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