Let P be a square matrix such that P² = I - P. For $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }, \mathrm{\delta }\in \mathrm{N}$ , if ${\mathrm{P}}^{\mathrm{\alpha }}+{\mathrm{P}}^{\mathrm{\beta }}=\mathrm{\gamma I}-29\mathrm{P} \mathrm{and} {\mathrm{P}}^{\mathrm{\alpha }}-{\mathrm{P}}^{\mathrm{\beta }}=\mathrm{\delta I}-13\mathrm{P}$, then $\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }-\mathrm{\delta }$ is equal to

$\begin{array}{l}{\mathrm{P}}^3={\mathrm{P}}^2\cdot \mathrm{P}=(\mathrm{I}-\mathrm{P})\mathrm{P}=\mathrm{P}-(\mathrm{I}-\mathrm{P})=2\mathrm{P}-\mathrm{I}\\ {\mathrm{P}}^4={\mathrm{P}}^2\cdot {\mathrm{P}}^2=(\mathrm{I}-\mathrm{P})(\mathrm{I}-\mathrm{P})=2\mathrm{I}-3\mathrm{P}\left({\mathrm{P}}_0\right)\end{array}$
Similarly ${\mathrm{P}}^5=5\mathrm{P}-3\mathrm{I},{\mathrm{P}}^6=5\mathrm{I}-8\mathrm{P},{\mathrm{P}}^7=13\mathrm{P}-8\mathrm{I},{\mathrm{P}}^8=13\mathrm{I}-21\mathrm{P}$
$\begin{array}{l}\Rightarrow {\mathrm{P}}^6+{\mathrm{P}}^8=18\mathrm{I}-29\mathrm{P}\\ {\mathrm{P}}^8-{\mathrm{P}}^6=-13\mathrm{P}+8\mathrm{I},\Rightarrow \mathrm{\alpha }=8,\mathrm{\beta }=6,\mathrm{\gamma }=18,\mathrm{\delta }=8\Rightarrow \mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }-\mathrm{\delta }=24\end{array}$