Construct a $2$ $\times$ $2$ matrix $A$ ($\neq$ $I$) with entries from $\mathbb{R}$ such that $A^3=I$.
First give me some hint. How to construct this kinds of matrices, is there any rule...
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1If I say "I have a transformation $T$ such that, if I do it three times, I go back to the way things were," what would you conjure up? – Henry Swanson Feb 06 '14 at 04:24
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think rotations – malxmusician212 Feb 06 '14 at 05:08
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Note that the eigenvalues of $A$ must satisfy $\lambda^3=1$ and if complex, must be complex conjugates.
One possibility: Eigenvalues are both $1$. This is not possible since $A$ is not the identity matrix and Jordon block of size 2 will not work.
The other possibility is $\lambda = -1/2 \pm i \sqrt{3}/2$. This gives $$A = \pmatrix{-{{1}\over{2}}&{{\sqrt{3}}\over{2}}\cr -{{\sqrt{3}}\over{2 }}&-{{1}\over{2}}\cr }$$
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