Word Count : 511

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The first Pauli matrix is [ \sigma_1 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}. ] We are asked to calculate [ U_1(\theta) = \exp(i \theta \sigma_1) = 1 + i \theta \sigma_1 - \frac{\theta^2}{2}\sigma_1^2 - i \frac{\theta^3}{6}\sigma_1^3 + \frac{\theta^4}{24}\sigma_1^4 + \dots ] and show that it is unitary and find its determinant. First, note the powers of (\sigma_1): [ \sigma_1^2 = \begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix} \begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix} = \begin{bmatrix}1 & 0 \ 0 & ___ ___ _____ _________ __________.
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