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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Matrices

NDA 2016 · Mathematics · Matrices

Consider the following in respect of the matrix A=( array cc -1 & 1...

Consider the following in respect of the matrix \mathrm{A}=\left(−111−1\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right) 1. A2=−A\quad \mathrm{A}^2=-\mathrm{A} 2. A3=4 A\quad \mathrm{A}^3=4 \mathrm{~A} Which of the above is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

B. 2 only

Step-by-step solution

A=[−111−1]A⋅A=[−111−1][−111−1]=[2−2−22]=−2[−111−1]A2=−2AA2⋅A=−2[−111−1][−111−1]=−2[2−2−22]=4[−111−1]A3=4A\begin{aligned} & A=\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right] \\ & A \cdot A=\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right]\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right] \\ & =\left[\begin{array}{cc} 2 & -2 \\ -2 & 2 \end{array}\right]=-2\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right] \\ & A^2=-2 A \\ & A^2 \cdot A=-2\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right]\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right] \\ & =-2\left[\begin{array}{cc} 2 & -2 \\ -2 & 2 \end{array}\right]=4\left[\begin{array}{cc} -1 & 1 \\ 1 & -1 \end{array}\right] \\ & A^3=4 A \end{aligned} Hence A2≠−A,A3=4AA^2 \neq-A, A^3=4 A
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