Suppose $a \neq 0$, and the n-by-n matrix $$A = \begin{pmatrix} a & 1 & 0 & ...\\ 0 & a & 1 & ...\\ & & ... \\ & & ... & a \end{pmatrix}$$ (that is, $a$ on the diagonal and $1$ on the super-diagonal), with $n>3$.
- Determine the minimal polynomial of $A$.
- Provide a simple formula for finding the inverse of $A$ in a systematic way. That is, provide the steps required to systematically determine $A^{-1}$.
I think that I have answered part 1: the minimal polynomial should be (I think) $m_{A}(s) = (s-a)^{n}$, since the size of the block is $n$. But I have no idea what process to use for the second part.