31. Use the details of the proof of Theorem 8.6.1 to find a lower triangular matrix R and an upper triangular matrix T such that A = RT where

$$

A = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ \end{bmatrix}

$$