TIFR Mathematics 2025 | Part A | Question: 17

Question

Let $n \geqslant 1$. Consider non-zero $n \times n$ real matrices $A$ such that $\operatorname{trace}(A X)=0$, for every real matrix $X$ with $\operatorname{trace}(X)=0$. Consider the following assertions.

1. For every such matrix $A, \operatorname{trace}(A)^{n}=n^{n} \cdot \operatorname{det}(A)$.
2. For every such matrix $A$, the rank of $A$ is $n$.

Which of the following sentences is correct?

• A. $\text{(i)}$ is correct and $\text{(ii)}$ is incorrect.
• B. $\text{(ii)}$ is correct and $\text{(i)}$ is incorrect.
• C. $\text{(i)}$ and $\text{(ii)}$ are correct.
• D. $\text{(i)}$ and $\text{(ii)}$ are incorrect.