GATE 2026 PH – Question 37
Consider the Pauli matrices $$\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ The value of $\operatorname{Tr}(\sigma_z[\sigma_x,\sigma_y])$ is
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Correct answer: (C) $4i$
Explanation
**Commutator of Pauli matrices:** $[\sigma_x,\sigma_y]=\sigma_x\sigma_y-\sigma_y\sigma_x=2i\sigma_z$.
(Check: $\sigma_x\sigma_y=i\sigma_z$ and $\sigma_y\sigma_x=-i\sigma_z$.)
**Trace:**
$$\text{Tr}\left(\sigma_z[\sigma_x,\sigma_y]\right)=\text{Tr}(\sigma_z\cdot2i\sigma_z)=2i\,\text{Tr}(\sigma_z^2)=2i\,\text{Tr}(I_{2\times2})=2i\times2=\mathbf{4i}.$$