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NDA 2019 Maths · Mathematics · Matrices · medium

The inverse of an invertible symmetric matrix is symmetric. 2. For any square matrix A, the matrix A-A^T is skew-symmetric.

Single correct option

Consider the following statements:
1. The inverse of an invertible symmetric matrix is symmetric.
2. For any square matrix AA, the matrix A−ATA-A^T is skew-symmetric.
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: (C) Both 1 and 2

Statement 1: from AT=AA^T=A we get (A−1)T=(AT)−1=A−1(A^{-1})^T=(A^T)^{-1}=A^{-1}, so the inverse is symmetric. Statement 2: (A−AT)T=AT−A=−(A−AT)(A-A^T)^T=A^T-A=-(A-A^T), so it is skew-symmetric. Both statements are correct. The sum A+ATA+A^T, by contrast, would be symmetric, which is a typical wrong swap.
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Original question written in the NDA 2019 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