Q188 / 270 · NDA 2017 (Phase 1) · Mathematics

If $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$, then the value of $A^4$ is

  • Correct answer
    A
    $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
  • B
    $\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$
  • C
    $\left[\begin{array}{ll}0 & 0 \\ 1 & 1\end{array}\right]$
  • D
    $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$

Solution

$\begin{aligned} & A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \\ & A^4=A^2 \cdot A^2 \\ & = \left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \cdot \left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \\ & =\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \cdot\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ & =\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\end{aligned}$

Hence, the correct answer is option: 'A'.

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