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NDA 2016 Maths · Mathematics · Matrices · easy

The inverse of an orthogonal matrix is its transpose. 2. If A and B are invertible matrices of the same order, then (AB)^-1=A^-1B^-1.

Single correct option

Consider the following statements:
1. The inverse of an orthogonal matrix is its transpose.
2. If AA and BB are invertible matrices of the same order, then (AB)−1=A−1B−1(AB)^{-1}=A^{-1}B^{-1}.
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

An orthogonal matrix satisfies AAT=IAA^T=I, so A−1=ATA^{-1}=A^T and statement 1 is true. The inverse of a product reverses the order: (AB)−1=B−1A−1(AB)^{-1}=B^{-1}A^{-1}, which equals A−1B−1A^{-1}B^{-1} only when the two commute. Statement 2 is therefore false.
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Original question written in the NDA 2016 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