Question

Using cayley-hamilton theorem,find a8 , if a = 1 2 2 1 .

Answer 1

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up vote6down voteaccepted

Given that P(t)=t4−2t2+1P(t)=t4−2t2+1, the Cayley-Hamilton Theorem yields that

P(A)=OP(A)=O

where OO is 4 by 4 zero matrix. Then

O=A4−2A2+I⇔A4=2A2−I⇒A8=(2A2−I)2O=A4−2A2+I⇔A4=2A2−I⇒A8=(2A2−I)2

EDIT: As suggested, this can be further simplified such that

A8=4A4−4A2+I=4(2A2−I)−4A2+I=4A2−3I

Answer 2

-4a2+|=4a2-3 is right answer