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TIFR Mathematics 2023 | Part B | Question: 4

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Answer whether the following statements are True or False.

Let $A, B \in \mathrm{M}_{2}(\mathbb{Z} / 2 \mathbb{Z})$ be such that $\operatorname{tr}(A)=\operatorname{tr}(B)$ and $\operatorname{tr}\left(A^{2}\right)=\operatorname{tr}\left(B^{2}\right)$. Then $A$ and $B$ have the same eigenvalues.

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FALSE

Let A=$\begin{pmatrix} 1 &0 \\ 0& 1 \end{pmatrix}$ and B=$\begin{pmatrix} 0 &0 \\ 0& 0 \end{pmatrix}$

now $\tr(A)=\tr(B)$ and $\tr(A^2)=\tr(B^2)$, But all the eigenvalues of B are 0 and 1 is an eigenvalue of A...
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