Show that if $A=L D L^{T}$, where $L \in \mathbb{R}^{m \times m}$ is a lower triangular matrix with all elements on the main diagonal being unity and $D \in \mathbb{R}^{m \times m}$ is a diagonal matrix with positive elements, then $A$ is positive definite. Find $L$ and the corresponding $D$ when

$$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 3 \end{array}\right]$$