Let $\mathrm{A}$ and $\mathrm{B}$ be any two $\mathrm{n} \times \mathrm{n}$ matrices such that the following conditions hold : $\mathrm{AB}=\mathrm{BA}$ and there exist positive integers $\mathrm{k}$ and $\ell$ such that $\mathrm{A}^{\mathrm{k}}=\mathrm{I}$ (the identity matrix) and $\mathrm{B}^{\prime}=0$ (the zero matrix). Then-

• A

  $\mathrm{A}+\mathrm{B}=\mathrm{I}$
• Correct answer
  B

  $\operatorname{det}(\mathrm{AB})=0$
• C

  $\operatorname{det}(\mathrm{A}+\mathrm{B}) \neq 0$
• D

  $(\mathrm{A}+\mathrm{B})^{\mathrm{m}}=0$ for some integer $\mathrm{m}$

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