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NDA 2018 Maths · Mathematics · Matrices · medium

The determinant of a skew-symmetric matrix of odd order is zero. 2. Every symmetric matrix is invertible.

Single correct option

Consider the following statements: 1. The determinant of a skew-symmetric matrix of odd order is zero. 2. Every symmetric matrix is invertible. Which of the statements given above 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: (A) 1 only

For skew-symmetric AA of order nn, ∣A∣=∣AT∣=∣−A∣=(−1)n∣A∣|A|=|A^T|=|-A|=(-1)^n|A|; for odd nn this forces ∣A∣=0|A|=0, so statement 1 is true. The zero matrix, or (1111)\begin{pmatrix}1&1\\1&1\end{pmatrix}, is symmetric yet singular, so statement 2 is false.
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Original question written in the NDA 2018 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