The system of simultaneous linear equations

$$ \begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 \\ & 2 x+3 y+z=6 \text { has } \end{aligned} $$

If $P$ is a non-singular matrix such that $I+P+P^2+\ldots \ldots+P^n=0(0$ denotes the null matrix $)$, then $P^{-1}=$

If $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$ and $\operatorname{det}\left(A^2\right)=25$, then $|\alpha|=$

$P$ is a $3 \times 3$ square matrix and $\operatorname{Tr}(P) \neq 0$. If $\operatorname{Tr}\left(P-P^I\right)+$ $\operatorname{Tr}\left(P+P^T\right)+\frac{\operatorname{Tr}(P)}{\operatorname{Tr}\left(P^T\right)}+\operatorname{Tr}(P) \times \operatorname{Tr}\left(P^T\right)=0$, then $\operatorname{Tr}(P)=$