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NDA 2025 Maths · Mathematics · Matrices · easy

For a square matrix A of order 3: 1. |2A|=2|A| 2. If A is skew-symmetric, then |A|=0 Which of the statements is/are correct?

Single correct option

Consider the following statements for a square matrix AA of order 33:
1. ∣2A∣=2∣A∣|2A|=2|A|
2. If AA is skew-symmetric, then ∣A∣=0|A|=0
Which of the statements is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

Each of the three rows gets multiplied by 22, so ∣2A∣=23∣A∣=8∣A∣|2A|=2^3|A|=8|A| and statement 1 is false. For skew-symmetric AA of odd order, ∣A∣=∣AT∣=∣−A∣=(−1)3∣A∣=−∣A∣|A|=|A^T|=|-A|=(-1)^3|A|=-|A|, hence ∣A∣=0|A|=0, so statement 2 is true.
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Original question written in the NDA 2025 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