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

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