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NDA 2 2025 Maths, question 15

Topic: Matrices and determinants

Q15

Consider the following in respect of non-singular matrices AA and BB :

I. (AB)−1=A−1B−1(AB)^{-1}=A^{-1}B^{-1}

II. (BA)(AB)−1=I(BA)(AB)^{-1}=I, where II is the identity matrix

III. (AB)T=ATBT(AB)^{T}=A^{T}B^{T}

How many of the above are correct?

  1. (a) NoneCorrect answer
  2. (b) One
  3. (c) Two
  4. (d) All three

Solution.

Answer: (a) None

  1. In general (AB)⁻¹=B⁻¹A⁻¹ and (AB)ᵀ=BᵀAᵀ: both operations reverse order.
  2. Also (BA)(AB)⁻¹=I would require BA=AB, which need not hold.
  3. A=[[1,1],[0,1]] and B=diag(1,2) are invertible noncommuting counterexamples to all three claimed universal identities.