The Exercise

The Pauli matrices $X$, $Y$ and $Z$ are the observables of a qubit along three perpendicular axes; for a spin $\tfrac12$ particle they are the spin components in units of $\hbar/2$. Exercise 2.41 asks to show that two different ones among them anticommute: reversing the order of their product reverses its sign.

The Pauli Matrices

The Pauli $X$ matrix is the complex $2 \times 2$ matrix $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$(1)^✦ The Pauli $Y$ matrix, with $i$ the imaginary unit, is $$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},$$(2) and the Pauli $Z$ matrix is the diagonal matrix $$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(3)

The Pauli matrices $\sigma_0, \sigma_1, \sigma_2, \sigma_3$ are the four complex $2 \times 2$ matrices $$\sigma_0 = I, \qquad \sigma_1 = X, \qquad \sigma_2 = Y, \qquad \sigma_3 = Z,$$(4)^✦ with $I$ the $2 \times 2$ identity matrix.

The Claim

What Is Checked

The claim is stated over the Pauli matrices (4), and through them over the three explicit matrices (1), (2) and (3); the products and the sum in (5) are those of complex $2 \times 2$ matrices.