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NDA 2020 Maths · Mathematics · Matrices · medium

The determinant of every skew-symmetric matrix of even order is zero. 2. Every square matrix can be written as the sum of a symmetric and a…

Single correct option

Consider the following statements:
1. The determinant of every skew-symmetric matrix of even order is zero.
2. Every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.
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

Skew-symmetry forces det⁡A=(−1)ndet⁡A\det A=(-1)^n\det A, so the determinant is zero only for odd order; [01−10]\begin{bmatrix}0&1\\-1&0\end{bmatrix} has determinant 1, so statement 1 is false. Statement 2 is true: A=12(A+AT)+12(A−AT)A=\tfrac12(A+A^T)+\tfrac12(A-A^T).
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Original question written in the NDA 2020 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