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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Matrices

NDA 2022 · Mathematics · Matrices

Consider the following in respect of the matrices: A= [ array ll m...

Consider the following in respect of the matrices: A=A=\left[mn\begin{array}{ll}m & n\end{array}\right], B=\left[−n−m\begin{array}{ll}-n & -m\end{array}\right] and C=\left[m−m\begin{array}{c}m \\ -m\end{array}\right] 1.C A=C B2.2.A C=B C3.3.C(A+B)=C A+C BWhichoftheabovestatementsis/arecorrect?Which of the above statements is/are correct?

Options

  1. A1 only
  2. B2 only
  3. C2 and 3
  4. D1 and 2

Correct answer

C. 2 and 3

Step-by-step solution

C A=\left[m−m\begin{array}{c}m \\ -m\end{array}\right]\left[mn\begin{array}{ll}m & n\end{array}\right]=\left[m2mn−m2−mn\begin{array}{cc}m^2 & m n \\ -m^2 & -m n\end{array}\right]C B=\left[m−m\begin{array}{c}m \\ -m\end{array}\right][-n-m]=\left[−mn−m2mnm2\begin{array}{cc}-m n & -m^2 \\ m n & m^2\end{array}\right]\therefore \quad C A \neq C B So, statement 1 is not correct. AC=[mn][m−m]=[m2−mn]BC=[−n−m][m−m]=[−mn+m2]\begin{aligned} & A C=\left[\begin{array}{ll}m & n\end{array}\right]\left[\begin{array}{c}m \\ -m\end{array}\right]=\left[m^2-m n\right] \\ & B C=\left[\begin{array}{ll}-n & -m\end{array}\right]\left[\begin{array}{c}m \\ -m\end{array}\right]=\left[-m n+m^2\right]\end{aligned}\therefore \quad A C=B CSo,statement2iscorrect.So, statement 2 is correct.C(A+B)=[m−m][m−nn−m]=[m2−mnmn−m2−m2+mn−mn+m2]CA+CB=[m2−mnmn−m2−m2+mn−mn+m2]=C(A+B)\begin{aligned} & C(A+B)=\left[\begin{array}{c}m \\ -m\end{array}\right]\left[\begin{array}{ll}m-n & n-m\end{array}\right] \\ & =\left[\begin{array}{cc}m^2-m n & m n-m^2 \\ -m^2+m n & -m n+m^2\end{array}\right] \\ & C A+C B=\left[\begin{array}{cc}m^2-m n & m n-m^2 \\ -m^2+m n & -m n+m^2\end{array}\right]=C(A+B)\end{aligned}$ So, statement 3 is correct.
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