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Mathematics Question on Definite Integral

Let
M=[0−α α0 ]M = \begin{bmatrix} 0 & -\alpha \\\ \alpha & 0 \\\ \end{bmatrix}
where α is a non-zero real number an
N=∑k=149M2k.N = \sum\limits_{k=1}^{49} M^{2k}. If (I−M2)N=−2I(I - M^2)N = -2I
then the positive integral value of α is ____ .

Answer

The correct answer is 1
M=[0−α α0 ],M2=[−α20 0−α2 ]=−α2IM = \begin{bmatrix} 0 & -\alpha \\\ \alpha & 0 \\\ \end{bmatrix},M^2 = \begin{bmatrix} -\alpha^2 & 0 \\\ 0 & -\alpha^2 \\\ \end{bmatrix} = -\alpha^2I
N=M2+M4+....+M98N = M² + M^4 + .... + M^{98}
=[−α2+α4−α6+....]I= [ -α² + α^4 - α^6 + .... ]I
=−α2(1−(−α2)49)1+α2.I= \frac{-α² (1-(-α²)^{49})}{ 1+α²} . I
I−M2=(1+α2)II - M² = ( 1 + α² )I
(I−M2)N=−α2(α98+1)=−2(I - M²)N = -α² (α^{98} + 1) = -2
Therefore α = 1