Let

$$A=\left[\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ 4 & 17 & 13 & 11 \\ 3 & 13 & 13 & 12 \\ 2 & 11 & 12 & \lambda \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 1 \\ 3 \\ 2 \end{array}\right]$$

where $\lambda$ is a real parameter. Find the $L U$ factorization of the matrix $A$. Give the constraint on $\lambda$ for A to be positive definite.

For $\lambda=18$, use this factorization to solve the system $A x=b$ via forward and backward substitution.