Exercise 7

Consider the matrix $$A = \left[\begin{array}{rrrr} a & a & a & a \\ a & b & b & b \\ a & b & c & c \\ a & b & c & d \\ \end{array}\right]\text{.}$$

1. Find the $LU$ factorization of $A$.

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