Let $\displaystyle M_n(\mathbb{C})$ be the space of all $\displaystyle n\times n$ matrices with complex entries.

Consider the matrices

$\displaystyle u=\left(\begin{array}{cccc}1 &0 &\cdots &0\\0 &q &\cdots &\vdots\\0 &0 &\ddots &\vdots\\0 &0 &\cdots &q^{n-1}\end{array}\right)$ and $\displaystyle v=\left(\begin{array}{ccccc} 0 &1 &0 &\cdots &0\\0 &0 &1 &\cdots &0\\0 &0 &\vdots &\ddots &0\\\vdots &\vdots &\cdots &\cdots &1\\1 &\cdots &0 &\cdots &0\end{array}\right)$

Where $\displaystyle u^n=v^n=1$ and $\displaystyle q^n=1$.

Show that the matrices $\displaystyle u \text{ and } v$ generate the whole of $\displaystyle M_n(\mathbb{C})$.

The two by two case is simple, since you can easily find four linearly independant matrices which you can use as the basis. However, I am convinced there has to be some easier way? For the n by n case we would then need to find $\displaystyle n^2$ linearly independant elements which is a little cumbersome.