Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 8D (Numerical Analysis)

Gaussian elimination without pivoting givesThus $\det A=3(\lambda -20)$ and the solution is unique exactly when $\lambda \ne 20$.

For $\lambda =20$, forward substitution in a triangular system in $Ly=b$ gives $y=(1,1,3,\mu -10{)}^T$. Consistency of $Ux=y$ requires $\boxed{\mu =10}$. backward substitution in a triangular system then gives