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Sums of powers of a matrix with A^2 = -k I

This type of question, asked 2 times: 25 Jun 2022, 29 Jun 2022. All 2 questions below, oldest first.

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  1. Question 1

    25 Jun 2022

    71
    Let A=[0−220]A =\begin{bmatrix} 0 & - 2 \\ 2 & 0 \end{bmatrix}. If MM and NN are two matrices given by M=∑k=110A2kM=\sum_{k= 1}^{10} A^{2k} and N=∑k=110A2k−1N=\sum_{k= 1}^{10} A^{2k- 1} then MN2MN^{2} is

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  2. Question 2

    29 Jun 2022

    179
    Let M=[0−αα0]M =\begin{bmatrix} 0 & - \alpha \\ \alpha & 0 \end{bmatrix}, where α\alpha is a non-zero real number an N=∑k=149M2kN =\sum_{k = 1}^{49} M^{2k}. If (I−M2)N=−2I\left( I -M^{2} \right)N = - 2I, then the positive integral value of α\alpha is