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Let be $\alpha∈R, n≥2$, and let be $A∈\textbf{X}_\textbf{n}\textbf{(R)}$ be a matrix satisfying the equation $$A^n+\frac{1}{4} α^2 A^{(n-2)}=αA^{(n-1)}$$ Prove that $A$ is similar to an upper triangular matrix and lower triangular matrix.

Thank you.

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  • $\begingroup$ Please try formatting your question with TeX as in math.stackexchange.com/help/notation it will make it far more readable. $\endgroup$ – Alex J Best May 3 at 10:07
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    $\begingroup$ I would start by factorising and after cancelling the A to n-2 you can solve as a quadratic $\endgroup$ – W M Seath May 3 at 10:12

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