Need help understanding how to properly setup the gaussian table. How/when do I use the last row ($\pi_1+\pi_2+\pi_3+\pi_4 =1$) of the systems? Does it appear as $\left[1\;1\;1\;1\;|\;1 \right] $from the beginning? Just need help with the first few steps to get me started.
Example $$ \left\lbrace \begin{array}{rcrcrcrcl} .7π_1 &+& 0 &+& 0 &+& .03π_4 &=& π_1 \\ .06π_1 &+& .7π_2 &+& .15π_3 &+& .03π_4 &=& π_2 \\ .18π_1 &+& .18π_2 &+& .7π_3 &+& .24π_4 &=& π_3 \\ .06π_1 &+& .12π_2 &+& .15π_3 &+& .7π_4 &=& π_4 \\ π_1 &+& π_2 &+& π_3 &+& π_4 &=& 1 \\ \end{array} \right. $$ Got the values from a transition matrix. Answer: $0.0311,0.2442,0.4139,0.3108$.
You can replace any of the four equations (e.g. last one) with the last equation $\pi_1+\cdots+ \pi_4=1$ to form the augmenterd matrix (note all $\pi$s were transfered to the left): $$ \left[ \begin{array}{cccc|c} -0.3&0&0&0.03&0\\ 0.06&-0.3&0.15&0.03&0\\ 0.18&0.18&-0.3&0.24&0\\ 1&1&1&1&1\\ \end{array} \right] $$ Can you solve it using the Gauss elimination method?