For the matrix

$$A=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 5 & 5 & 5 \\ 1 & 5 & 14 & 14 \\ 1 & 5 & 14 & \lambda \end{array}\right]$$

find a factorization of the form

$$A=L D L^{\top} \text {, }$$

where $D$ is diagonal and $L$ is lower triangular with ones on its diagonal.

For what values of $\lambda$ is $A$ positive definite?

In the case $\lambda=30$ find the Cholesky factorization of $A$.