Given that
$\sum_{n}^{\infty} \mu^{n} M^{n} = (I-\mu M)^{-1}$
Wherein $\mu$ is a scalar, $M$ is a matrix and $I$ is the identity matrix of the same dimension as $M$. How do I use this to find the n-th power of the matrix $M$?
The exercise states
Conclude that $M^{n}$ can be obtained as the coefficient of $\mu^{n}$ in the power expansion of $(I-\mu M)^{-1}$.
But what are the arguments for this conclusion?
Supposing convergence, $$ (I-\mu M)\sum_{n=0}^{\infty}\mu^{n}M^{n} = \sum_{n=0}^{\infty}\mu^{n}M^{n} - \sum_{n=0}^{\infty}\mu^{n+1}M^{n+1} =\sum_{n=0}^{\infty}\mu^{n}M^{n}-\sum_{n=1}^{\infty}\mu^{n}M^{n} = I. $$ As I know, the formula isn't terribly useful for calculating $M^n$ in the practice.