Chapter 10, Problem 2P

(a) Use naive Gauss elimination to decompose the following system according to the description in Sec. 10.1.2.

$\begin{array}{l}10x_1+2x_2-x_3=27\\ -3x_1-6x_2+2x_3=-61.5\\ x_1+x_2+5x_3=-21.5\end{array}$

Then, multiply the resulting $\left[L\right]$ and $\left[U\right]$ matrices to determine that $\left[A\right]$ is produced. (b) Use LU decomposition to solve the system. Show all the steps in the computation. (c) Also solve the system for an alternative right-hand-side vector: ${\left\{B\right\}}^T=\left[1218-6\right]$.