$$A=\left[\begin{array}{cccc} 1 & 2 & 1 & 2 \\ 2 & 5 & 5 & 6 \\ 1 & 5 & 13 & 14 \\ 2 & 6 & 14 & \lambda \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 3 \\ 7 \\ \mu \end{array}\right]$$

where $\lambda$ and $\mu$ are real parameters. Find the $L U$ factorisation of the matrix $A$. For what values of $\lambda$ does the equation $A x=b$ have a unique solution for $x$ ?

For $\lambda=20$, use the $L U$ decomposition with forward and backward substitution to determine a value for $\mu$ for which a solution to $A x=b$ exists. Find the most general solution to the equation in this case.