The determinant of the matrix \(P = \begin{pmatrix} 1 & 0 & -1 \\ 3 & 4 & 5 \\ -1 & 0 & 1 \end{pmatrix}\) is calculated as follows: \(|P| = 1 \cdot (4 \cdot 1 - 5 \cdot 0) - 0 \cdot (3 \cdot 1 - 5 \cdot -1) + (-1) \cdot (3 \cdot 0 - 4 \cdot -1) = 4 - 0 - 4 = 0\) Thus, the determinant of the matrix \( P \) is \(|P| = 0\).