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As stated. I'm not sure where to start. I've been looking at matrices if the determinant = 1,-1, but that has been clearly the wrong way to go about this. Any tips?

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    $\begingroup$ Try a rotation matrix $\endgroup$ – Dark Malthorp Apr 2 at 23:31
  • $\begingroup$ make a matrix that means a rotation of $90^\circ$ in the plane $\endgroup$ – Will Jagy Apr 2 at 23:31
  • $\begingroup$ Well, you're looking in the right place -- the matrix you seek must have determinant $\pm 1$. But try thinking of matrices as representing transformations --- is there some operation you can do four times and have it be the same as doing nothing? $\endgroup$ – John Hughes Apr 2 at 23:31
  • $\begingroup$ OH wow. Thanks a ton. Sometimes you get so caught up in the semantics and equations, you forget to just visualize it sometimes. :) $\endgroup$ – Steve Apr 3 at 0:03
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Hint: Think about rotation matrices.

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Consider the diagonal matrix $$ D= \left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] $$ Note that $D^4=I$.

Now, if $P$ is invertible, then $A=PDP^{-1}$ satisfies $$ A^4=PD^4P^{-1}=PIP^{-1}=PP^{-1}=I $$ For instance, taking $$ P=\left[\begin{array}{rr} 1 & 5 \\ 2 & 11 \end{array}\right] $$ gives $$ A =\overset{P}{\left[\begin{array}{rr} 1 & 5 \\ 2 & 11 \end{array}\right]}\overset{D}{\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]}\overset{P^{-1}}{\left[\begin{array}{rr} 11 & -5 \\ -2 & 1 \end{array}\right]}=\left[\begin{array}{rr} 21 & -10 \\ 44 & -21 \end{array}\right] $$

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