Exercise 5, § 10

Let $\mathbf{A}$ be a ring, $\mathbf{L}, \mathbf{M}, \mathbf{N}$ any three indexing sets $U = (a_{\lambda \mu})$ a matrix of $\mathbf{A}^{\mathbf{L} \times \mathbf{M}}$ and $V = (b_{\mu \nu})$ a matrix of $\mathbf{A}^{\mathbf{M} \times \mathbf{N}}$; if, for every ordered pair $(\lambda, \nu) \in \mathbf{L} \times \mathbf{N}$, the family $(a_{\lambda \mu} b_{\mu \nu})$ ($\mu \in \mathbf{M}$) has finite support, the element $c_{\lambda \nu} = \sum_{\mu} a_{\lambda \mu} b_{\mu \nu}$ is defined and it is also said that the matrix $(c_{\lambda \nu})$ is the product $UV$ of $U$ by $V$. When the products $UV'$ and $UV''$ are defined, so is $U(V' + V'')$ and $UV' + UV'' = U(V' + V'')$; is the converse true? Give an example of three infinite matrices, $U, V, W$ such that the products $UV, VW, U(VW)$ and $(UV)W$ are defined but $U(VW) \neq (UV)W$ (take $U$ to be a matrix with one row all of whose elements are equal to 1, $W$ its transpose and $V$ a matrix all of whose elements are 0, 1 or $-1$ and which has only a finite number of elements $\neq 0$ in each row and each column. Determine by induction the elements of $V$ such that $UV = 0$ but $VW$ has a single element $\neq 0$.)