Let $A$ be a $2 \times 2$ matrix such that
$A^2+A+\left(\begin{array}{ll}<br/>0 & 1 \\<br/>1 & 0<br/>\end{array}\right)=\left(\begin{array}{ll}<br/>0 & 0 \\<br/>0 & 0<br/>\end{array}\right)$
Let $I$ denote the $2 \times 2$ identity matrix. Which of the following statements is correct?

• Correct answer
  A

  Both $A$ and $A+I$ are invertible.
• B

  $A$ is invertible but $A+I$ may not be invertible.
• C

  Neither $A+I$ nor $A$ may be invertible.
• D

  $A+I$ is invertible but $A$ may not be invertible.

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