Question

Let $M =\begin{bmatrix}0 & -\alpha \\ \alpha & 0\end{bmatrix}$, where $\alpha$ is a non-zero real number an $N =\displaystyle\sum_{ k =1}^{49} M ^{2 k }$. If $\left( I - M ^{2}\right) N =-2 I$, then the positive integral value of $\alpha$ is______

Answer 1

$M =\begin{bmatrix}0 & -\alpha \\ \alpha & 0\end{bmatrix} ; M ^{2}=\begin{bmatrix}-\alpha^{2} & 0 \\ 0 & -\alpha^{2}\end{bmatrix}=-\alpha^{2} I$
$N = M ^{2}+ M ^{4}+\ldots \ldots+ M ^{98}=\left[-\alpha^{2}+\alpha^{4}-\alpha^{6}+\ldots\right] I$
$=-\alpha^{2} \frac{\left(1-\left(-\alpha^{2}\right)^{49}\right)}{1+\alpha^{2}} . I$
$I - M ^{2}=\left(1+\alpha^{2}\right) I$
$\left( I - M ^{2}\right) N =-\alpha^{2}\left(\alpha^{98}+1\right)=-2$
$\alpha=1$