###### Exercise 7

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*}

Find the \(LU\) factorization of \(A\text{.}\)

What conditions on \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) guarantee that \(A\) is invertible?