Prove the identity ((sigma cdot boldsymbol{p})^{2}=mathrm{I}^{(2)} p^{2}), where (sigma=left(sigma_{1}, sigma_{2}, sigma_{3} ight)) are the Pauli matrices, (boldsymbol{p})

Prove the identity \((\sigma \cdot \boldsymbol{p})^{2}=\mathrm{I}^{(2)} p^{2}\), where \(\sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)\) are the Pauli matrices, \(\boldsymbol{p}\) is the 3-momentum, \(\boldsymbol{p}=|\boldsymbol{p}|\), and \(\mathrm{I}^{(2)}\) is the unit \(2 \times 2\) matrix.