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All the four entries of the $2 \times 2$ matrix $\mathbf{P}=\left[\begin{array}{ll}p_{11} & p_{12} \\ p_{21} & p_{22}\end{array}\right]$ are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?

1. $p_{11} p_{22}-p_{12} p_{21}=1$
2. $p_{11} p_{22}-p_{12} p_{21}=-1$
3. $p_{11} p_{22}-p_{12} p_{21}=0$
4. $p_{11} p_{22}+p_{12} p_{21}=0$