Let $\alpha$ be a complex number such that $\alpha \neq 1$ and $\alpha^5=1$. Let $A=\left(\begin{array}{lll}0 & 0 & \alpha \\ 0 & \alpha & 0 \\ \alpha & 0 & 0\end{array}\right)$ and $I$ denote the identity matrix. Then the value of $I+A+A^2+A^3+A^4$ is

• Correct answer
  A

  $\left(1+\alpha^2+\alpha^4\right)\left(\begin{array}{ccc}1 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 1\end{array}\right)$
• B

  $\alpha\left(1+\alpha^2\right)\left(\begin{array}{ccc}1 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 1\end{array}\right)$
• C

  $\left(1+\alpha^2+\alpha^4\right)\left(\begin{array}{ccc}-1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & -1\end{array}\right)$
• D

  $\left(1+\alpha^2+\alpha^4\right)\left(\begin{array}{ccc}-1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & -1\end{array}\right)$

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