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NDA 2 2024 Maths, question 19

Topic: Matrices and determinants

Q19

Consider the following in respect of the matrices

P=[0c−b−c0ab−a0]P=\begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} and Q=[a2abacabb2bcacbcc2]Q=\begin{bmatrix} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{bmatrix}

I. PQPQ is a null matrix.

II. QPQP is an identity matrix of order 33.

III. PQ=QPPQ=QP

Which of the above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) I and IIICorrect answer
  4. (d) II and III

Solution.

Answer: (c) I and III

  1. Let v=(a,b,c)ᵀ, so Q=vvᵀ.
  2. Direct multiplication gives Pv=0.
  3. Since Pᵀ=−P, vᵀP=0 too.
  4. Consequently PQ=(Pv)vᵀ=0 and QP=v(vᵀP)=0.
  5. I and III are true; II is false.