TIFR ECE 2020 | Question: 9

Question

Let $A$ be an $n \times n$ matrix with the the property that $A^{m}=0$ for some $m \in \mathbb{N}$. Consider the following statements:

1. At least one entry of $A$ is zero
2. All eigenvalues of $A$ are zero
3. All diagonal entries of $A$ are zero

Which of the following is $\text{TRUE?}$

• A. Only statement $1$ is correct
• B. Statement $2$ alone is correct 
• C. Only statement $3$ is correct
• D. Only statements $1$ and $2$ are correct
• E. Only statements $2$ and $3$ are correct