spin projector in inverted matrix

Question

The following matrix $A$ is, \begin{equation} A= \begin{bmatrix} a+b-\sigma\cdot \textbf p & -x_1 \\ x_2 & a-b-\sigma\cdot \textbf p \end{bmatrix} \end{equation} The inversion of matrix $A$ is, \begin{equation} A^{-1}= \frac{\begin{bmatrix} a-b-\sigma\cdot \textbf p & x_1 \\ -x_2 & a+b-\sigma\cdot \textbf p \end{bmatrix}}{a^2-b^2+p^2-2\sigma\cdot \textbf p-x_1x_2} \end{equation} However, p.4 equation (31) of the paper where matrix $A$ is taken from shows the formula in different form, \begin{equation} A^{-1}= \frac{0.5\begin{bmatrix} (\sigma_o+\sigma\cdot \hat{\textbf p})(a-b+p) & (\sigma_o+\sigma\cdot \hat{\textbf p}) x_1 \\ -x_2(\sigma_o+\sigma\cdot \hat{\textbf p}) & \sigma_y(\sigma_o+\sigma\cdot \hat{\textbf p})\sigma_y(a+b-p) \end{bmatrix}}{a^2-b^2-p^2+2bp-x_1x_2} \end{equation} and $\hat{p}=\frac{p}{b}$ and $\sigma$'s are 2x2 pauli matrices such that $\sigma\sigma=1$. The $\sigma$s inside brackets are projectors. How it is derived? How projector is used to get the inversion of matrix $A$ that looks different from conventional method used to calculate $A^{-1}$? What kind of algebra it is using to get this?