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**Paul616** So I trying to find a 2 * 2 Matrix M where $\displaystyle M^n = I$ but $\displaystyle M^k$ where $\displaystyle k<n$ does not = I

So for n = 1 M=I,

for n = 2 $\displaystyle M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)$

for n = 3 $\displaystyle M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)$

for n = 4 $\displaystyle M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)$

These are just a few examples I found using algebraic methods however I can not find an example for n = 5, is there a possible reason for this?

Futhermore can you find a 2 * 2 matrix M where $\displaystyle M^5 = I$ and $\displaystyle M^{11} = 0$ ?

Also is there a better way to attempting these problems than working through several simultaneous equations?

Thanks