Exercise 2, § 10

(a) Let $X$ be a square matrix over a ring $\mathbf{A}$, which can be written as an upper triangular matrix $(X_{ij})$ of matrices ($1 \leq i \leq p,\ 1 \leq j \leq p$). Show that if each of the square matrices $X_{ii}$ ($1 \leq i \leq p$) is invertible, so is $X$ and $X^{-1}$ can be written as an upper triangular matrix $(Y_{ij})$ ($1 \leq i \leq p,\ 1 \leq j \leq p$) corresponding to the same partition of the indexing set. When $\mathbf{A}$ is a field, prove that this sufficient condition for $X$ to be invertible is also necessary.

(b) Give an example of a ring $\mathbf{A}$ and a matrix $\begin{pmatrix} a & 1 \\ 0 & b \end{pmatrix}$ (with $a \in \mathbf{A},\ b \in \mathbf{A}$) which is invertible without either $a$ or $b$ being invertible in $\mathbf{A}$ and whose inverse $\begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix}$ is such that $b' \neq 0$ and $c' \neq 0$ (take $\mathbf{A}$ to be the endomorphism ring of an infinite-dimensional vector space).