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If $A$ is a $3\times3$ invertible matrix over $\mathbb{R}$ such that $\det(A)=1$ and $\mathrm{tr}(A)=\mathrm{tr}(A^2)=0$, then

  A) $A^{3}=A+I$
  B) $A^{3}=I$
  C) $A^{2}=I$
  D) $A^{3}=A^{2}+I$

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Do you know the Cayley Hamilton theorem? What are the coefficients in the characteristic polynomial, and how are they related to trace? –  user27126 Jan 24 '13 at 21:50
Please give some thoughts on that yourself and not entirely copy your multiple choice question here. –  mez Jan 24 '13 at 21:52

1 Answer 1

up vote 5 down vote accepted

The statement of the Cayley–Hamilton theorem can be written as: $$ p(A) = A^{3} - tr{(A)}A^2 + \frac{1}{2} \left((tr{(A)}^2)-tr{(A)^2}\right)A -\det{(A)}I_3 = 0 $$ where the right-hand side designates a $3×3$ matrix with all entries reduced to zero.

Substituting known values, we get that $A^3$ = $\det{(A)}I_3 = I_3 $,
So $(B)$ is true.

I would suggest that you read: Caley-Hamilton Theorem

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thanks a lot Rustyn –  rese Jan 24 '13 at 22:36

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