Write the value of the determinant

$\left|\begin{array}{cc}p & p+1 \\ p-1 & p\end{array}\right|=p^{2}-(p+1)(p-1)$

$=p^{2}-\left(p^{2}-1\right)$

$=p^{2}-p^{2}+1$

$=1$

Hence, the value of the determinant $\left|\begin{array}{cc}p & p+1 \\ p-1 & p\end{array}\right|$ is 1 .