Let $\mathrm{A}$ be a matrix of order 5 such that $\mathrm{A}.\mathrm{M}.(-2)=2$ , $\mathrm{A}.\mathrm{M}.\left(3\right)=1$ and $\mathrm{A}.\mathrm{M}.\left(4\right)=2$ , then $\mathrm{trace}\left(\mathrm{A}\right)=$ _____.

Let $\mathrm{P}=\left[\begin{array}{ccc}5& 4& 0\\ 0& -2& 0\\ 0& 0& 1\end{array}\right]$ . Then $\mathrm{trace}({\mathrm{P}}^3+3{\mathrm{I}}_3)=$ _____.

If $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ -1& 3\end{array}\right]$ . Then ${\mathrm{A}}^5-3{\mathrm{A}}^4+{\mathrm{A}}^3-7{\mathrm{A}}^2+5\mathrm{A}+\mathrm{I}$ is equals to _____.

The system of simultaneous linear equations  $\mathrm{x}+2\mathrm{y}+3\mathrm{z}=5, \mathrm{x}+\mathrm{y}+4\mathrm{z}=10, 2\mathrm{x}+2\mathrm{y}+8\mathrm{z}=\mathrm{k}$ will have infinitely many solutions for $\mathrm{k}$ equals to _____.