##### Exercise7

Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrrr} a \amp a \amp a \amp a \\ a \amp b \amp b \amp b \\ a \amp b \amp c \amp c \\ a \amp b \amp c \amp d \\ \end{array}\right]\text{.} \end{equation*}
1. Find the $$LU$$ factorization of $$A\text{.}$$

2. What conditions on $$a\text{,}$$ $$b\text{,}$$ $$c\text{,}$$ and $$d$$ guarantee that $$A$$ is invertible?

in-context