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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Matrices

NDA 2024 · Mathematics · Matrices

Consider the following in respect of the matrices P=[ array ccc 0 &...

Consider the following in respect of the matrices \mathrm{P}=\left[0c−b−c0ab−a0\begin{array}{ccc} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{array}\right] \text{\text{ and } } \mathrm{Q}=\left[a2abacabb2bcacbcc2\begin{array}{ccc} a^2 & a b & a c \\ a b & b^2 & b c \\ a c & b c & c^2 \end{array}\right] A. PQ is null a matrix. B. QP is an identity matrix of order 3. C. PQ=\mathrm{PQ}=QP\mathrm{QP}whichoftheaboveis/arecorrect?which of the above is/are correct?

Options

  1. AA only
  2. BB only
  3. CA and C
  4. DB and C

Correct answer

C. A and C

Step-by-step solution

Wehavegiven,We have given,P=[0c−b−c0ab−a0],Q=[a2abacabb2bcacbcc2]⇒PQ=[000000000] and QP=[000000000]⇒PQ=QP⇒ option (c) is correct. \begin{aligned} & \mathrm{P}=\left[\begin{array}{ccc}0 & \mathrm{c} & -\mathrm{b} \\ -\mathrm{c} & 0 & \mathrm{a} \\ \mathrm{b} & -\mathrm{a} & 0\end{array}\right], \mathrm{Q}=\left[\begin{array}{ccc}\mathrm{a}^2 & \mathrm{ab} & \mathrm{ac} \\ \mathrm{ab} & \mathrm{b}^2 & \mathrm{bc} \\ \mathrm{ac} & \mathrm{bc} & \mathrm{c}^2\end{array}\right] \\ & \Rightarrow \mathrm{PQ}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] \text{ and } \mathrm{QP}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] \\ & \Rightarrow \mathrm{PQ}=\mathrm{QP} \Rightarrow \text{ option }(\mathrm{c}) \text{ is correct. }\end{aligned}$
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