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Consider the matrices A=\left(\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & 0 & -1 \\

0 & 0 & 0 \\ -1 & 0 & 1 \end{array}\right), \quad C=\left(\begin{array}{ccc} 1 & -2 & 1 \\ -2 & 4 & -2 \\ 1 & -2 & 1 \end{array}\right) M=\left(\begin{array}{ccc} 1+\alpha & 1 & 1-\alpha \\ 1 & 1 & 1 \\ 1-\alpha & 1 & 1+\alpha \end{array}\right), \alpha \in \mathbf{R} You aregiven that the matrices A, B and C satisfy АВ — ВА — АС — СА — ВС — СВ — 0, А? — ЗА, В? — 2B, С? — 6C. Find an invertible matrix P and a diagonal matrix D such that M = PDP-1. Show all working.

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