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NDA 1 2024 Maths, question 12

Topic: Matrices and determinants

Q12

Consider the following statements in respect of a non-singular matrix AA of order nn :

  1. A(adjAT)=A(adjA)TA(\text{adj}A^{T})=A(\text{adj}A)^{T}

  2. If A2=AA^{2}=A, then AA is identity matrix of order nn

  3. If A3=AA^{3}=A, then AA is identity matrix of order nn

Which of the statements given above are correct ?

  1. (a) 1 and 2 onlyCorrect answer
  2. (b) 2 and 3 only
  3. (c) 1 and 3 only
  4. (d) 1, 2 and 3

Solution.

Answer: (a) 1 and 2 only

  1. Adjugation commutes with transpose, so adj(Aᵀ)=(adj A)ᵀ and statement 1 is true.
  2. Multiplying A²=A by A⁻¹ gives A=I, proving statement 2.
  3. But A³=A gives only A²=I; A=−I is a nonsingular counterexample to statement 3.
  4. Hence 1 and 2 only.