1. ## Raising a Matrix to a power, help please!

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

2. Originally Posted by 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
Simultaneous equations probably best.

Have you looked at 3x3, 4x4, and 5x5 arrays for the similar identities?

3. Also, regarding the second question -- if you have $\displaystyle M^5 = I$ then $\displaystyle M^{11} = M^{10} \cdot M^1 = (M^5)^2 \cdot M = I^2 \cdot M = M \neq 0$

So clearly it isn't possible.

4. Originally Posted by 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?
for any positive integers $\displaystyle m,n$ you can always find an $\displaystyle m \times m$ matrix $\displaystyle M,$ over $\displaystyle \mathbb{C},$ such that $\displaystyle M^n = I$ but $\displaystyle M^k \neq I$ for all $\displaystyle 0 < k < n.$ an example is $\displaystyle M=\rho I,$ where $\displaystyle \rho$ is a primitive $\displaystyle n$th root of unity.