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TIFR Mathematics 2022 | Part A | Question: 9

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What is the largest value of $n$ for which there exists a set $\left\{A_{1}, \ldots, A_{n}\right\}$ of (distinct) nonzero matrices in $\mathrm{M}_{2}(\mathbb{C})$ such that $A_{i}^{*} A_{j}$ has trace zero for all $1 \leq i < j \leq n?$

  1. $1$
  2. Greater than $1$ but at most $4$
  3. Greater than $4$ but finite
  4. $\infty$
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