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If matrix a satisfies the equation A^(2)...

If matrix a satisfies the equation `A^(2)=A^(-1)`, then prove that `A^(2^(n))=A^(2^((n-1))), n in N`.

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`A^(2^(n))=A^(2.2^(n-1))=(A^(2))^(2^(n-1))`
`=(A^(-1))^(2^(n-1))=(A^(2^(n-1)))^(-1)=(A^(2.2^(n-1)))^(-1)`
`=((A^(2))^(2^(n-1)))^(-1)=((A^(-1))^(-1))^(2^((n-2)))=A^(2^((n-2)))`
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