= Solution
Define the <characteristic polynomial> using $p(\lambda)=\det(\lambda I-P)$, where $I$ is the <identity matrix>. Expanding the <determinant> along the first row gives
$$
\begin{aligned}
p(\lambda)&=(\lambda-0.6)\bigl((\lambda-0.8)(\lambda-0.7)-0.02\bigr)\\
&\quad+0.1\bigl(-0.1(\lambda-0.7)-0.06\bigr)-0.1\bigl(0.01+0.3(\lambda-0.8)\bigr)\\
&=\lambda^3-2.1\lambda^2+1.4\lambda-0.3\\
&=(\lambda-1)(\lambda-0.6)(\lambda-0.5).
\end{aligned}
$$
Thus the <characteristic equation> is
$$
\boxed{p(\lambda)=\lambda^3-\frac{21}{10}\lambda^2+\frac75\lambda-\frac3{10}=0.}
$$
The convention $\det(P-\lambda I)$ changes the sign of this cubic <polynomial> but gives the same <roots>.
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