The characteristics equation is ∣A−λI∣=0
∣∣3−λ−12λ∣∣=0−λ(3−λ)+2=0−3λ+λ2+2=0λ2−3λ+2=0
By Cayley-Hamilton Theorem,
A2−3A+2I=0⟹A2=3A−2I
A7=A4.A3A7=(A2)2.A2.AA7=(3A−2I)2.(3A−2I).AA7=(9A2−12A+4I)(3A2−2A)A7=(9(3A−2I)−12A+4I)(3(3A−3I)−2A)A7=(27A−18I−12A+4I)(9A−6I−2A)A7=(15A+4I)(7A−6I)A7=(105A2−188A+84I)A7=(105(3A−2I)−188A+84I)A7=315A−210A−188A+84IA7=127A−126IA7=127(3−120)−126(1001)A7=(381−1272540)−(12600126)A7=(255−127254−126)
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