Let M = [ cc 0 & - & 0 ], where is a non-zero real number an N = … - Vidyadip

Question

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

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Answer

d
$M =\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right] ; M ^{2}=\left[\begin{array}{cc}-\alpha^{2} & 0 \\ 0 & -\alpha^{2}\end{array}\right]=-\alpha^{2} I$

$N = M ^{2}+ M ^{4}+\ldots \ldots+ M ^{98}=\left[-\alpha^{2}+\alpha^{4}-\alpha^{6}+\ldots .\right] I$

$=-\alpha^{2} \frac{\left(1-\left(-\alpha^{2}\right)^{49}\right)}{1+\alpha^{2}} . I$

$I - M ^{2}=\left(1+\alpha^{2}\right) I$

$\left( I - M ^{2}\right) N =-\alpha^{2}\left(\alpha^{98}+1\right)=-2$

$\alpha=1$

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