Raising a Matrix to a power, help please!

**Paul616** · September 7th 2009, 07:06 AM

So I trying to find a 2 * 2 Matrix M where $M^n = I$ but $M^k$ where $k<n$ does not = I
So for n = 1 M=I,
for n = 2 $M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)$
for n = 3 $M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)$
for n = 4 $M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)$
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 $M^5 = I$ and $M^{11} = 0$ ?
Also is there a better way to attempting these problems than working through several simultaneous equations?
Thanks

**aidan** · September 7th 2009, 10:22 AM

Originally Posted by Paul616

So I trying to find a 2 * 2 Matrix M where $M^n = I$ but $M^k$ where $k<n$ does not = I
So for n = 1 M=I,
for n = 2 $M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)$
for n = 3 $M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)$
for n = 4 $M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)$
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 $M^5 = I$ and $M^{11} = 0$ ?
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?

**Defunkt** · September 7th 2009, 10:27 AM

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

So clearly it isn't possible.

**NonCommAlg** · September 7th 2009, 03:07 PM

Originally Posted by Paul616

So I trying to find a 2 * 2 Matrix M where $M^n = I$ but $M^k$ where $k<n$ does not = I
So for n = 1 M=I,
for n = 2 $M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)$
for n = 3 $M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)$
for n = 4 $M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)$
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 $m,n$ you can always find an $m \times m$ matrix $M,$ over $\mathbb{C},$ such that $M^n = I$ but $M^k \neq I$ for all $0 < k < n.$ an example is $M=\rho I,$ where $\rho$ is a primitive $n$ th root of unity.

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