Let $\mathrm{M}=\left[\begin{array}{cc}0& -\mathrm{\alpha }\\ \mathrm{\alpha }& 0\end{array}\right]$, where α is a non-zero real number an $\mathrm{N}=\sum _{\mathrm{k}=1}^{49} {\mathrm{M}}^{2\mathrm{k}}$. If (I - M²)N = -2I, then the positive integral value of α is ____.

$\begin{array}{l}\mathrm{M}=\left[\begin{array}{cc}0& -\mathrm{\alpha }\\ \mathrm{\alpha }& 0\end{array}\right];{\mathrm{M}}^2=\left[\begin{array}{cc}-{\mathrm{\alpha }}^2& 0\\ 0& -{\mathrm{\alpha }}^2\end{array}\right]=-{\mathrm{\alpha }}^2\mathrm{I}\\ \mathrm{N}={\mathrm{M}}^2+{\mathrm{M}}^4+\dots \dots +{\mathrm{M}}^{98}=\left[-{\mathrm{\alpha }}^2+{\mathrm{\alpha }}^4-{\mathrm{\alpha }}^6+\dots \right]\mathrm{I}\\ =-{\mathrm{\alpha }}^2\frac{\left(1-{\left(-{\mathrm{\alpha }}^2\right)}^{49}\right)}{1+{\mathrm{\alpha }}^2}.\mathrm{I}\\ \mathrm{I}-{\mathrm{M}}^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}-{\mathrm{\alpha }}^2& 0\\ 0& -{\mathrm{\alpha }}^2\end{array}\right]=\left(1+{\mathrm{\alpha }}^2\right)\mathrm{I}\\ \left(\mathrm{I}-{\mathrm{M}}^2\right)\mathrm{N}=\left(1+{\mathrm{\alpha }}^2\right)\mathrm{I}\left(\frac{-{\mathrm{\alpha }}^2\left({\mathrm{\alpha }}^{98}+1\right)}{1+{\mathrm{\alpha }}^2}\right).\mathrm{I}=-2I\\ \Rightarrow -{\mathrm{\alpha }}^2\left({\mathrm{\alpha }}^{98}+1\right)=-2\\ \mathrm{\alpha }=1\end{array}$