If  $P$ is a non null matrix of order  $3\times 3$ with real entries such that  $P^3=O$ then (consider  $O$ as  $3\times 3$ null matrix and  $I$ as  $3\times 3$ identity matrix)

$8P^3+I=(2P+I)(4P^2-2P+I)=I$

Taking determinant  $|2P+I|\ne 0,|4P^2-2P+I|\ne 0$  
Again  $I-8P^3=(I-2P)(4P^2+2P+I)=I$
Taking determinant  $|I-2P|\ne 0$
So  $|I+2P||I-2P|\ne 0$
$\Rightarrow |I-4P^2|\ne 0$
If  $P$ has integral entries  $|I-2P|$ and  $|I+2P|$ can be 1 or – 1. So $\left||I-4P^2|\right|=1.$