Solve linear system with 9 equations and 9 uknowns using Gauss-elimination

Question

I'm having difficulties doing this exercise.

$$-4T_1+T_2+T_4=-100$$ $$T_1-4T_2+T_3+T_5=-20$$ $$T_2-4T_3+T_6=-20$$ $$T_1-4T_4+T_5+T_7=-80$$ $$T_2+T_4-4T_5+T_6+T_8=0$$ $$T_3+T_5-4T_6+T_9=0$$ $$T_4-4T_7+T_8=-260$$ $$T_5+T_7-4T_8+T_9=-180$$ $$T_6+T_8-4T_9=-180$$

Any help is highly appreciated. It needs to be done using the Gaussian method, and im a little bit rusty and dont see the obvious/best way to go.

Sorry bout late upload of system. Was formatting the post.

Answer 1

Have you learned the Cramer's rule? You can compute the value of $\,T_i, i=1,...,9$ directly by using it.

I think you can write the system into a $9$x$10$ matrix and try some row or column transformations to make it close to ($I_9$,b), where $I_9$ is the $9$x$9$ identity matrix and b is a $9$x$1$ column vector.

I want to write all these as a comment of this problem, but my reputation is not enough to do that. So I write it as a answer. But it is just my thinking.

Hope my words can help you.

Answer 2

HINT.-Your (equivalent) matrix equation is $$\begin{equation}\begin{bmatrix} -4&1&0&1&0&0&0&0&0\\[1em]1&-4&1&0&1&0&0&0&0 \\[1em]0&1&-4&0&0&1&0&0&0 \\[1em]1&0&0&-4&1&0&1&0&0\\[1em]0&1&0&1&-4&1&0&1&0\\[1em]0&0&1&0&1&-4&0&0&1\\[1em]0&0&0&1&0&0&-4&1&0\\[1em]0&0&0&0&1&0&1&-4&1\\[1em]0&0&0&0&0&1&0&1&-4 \end{bmatrix}*\begin{bmatrix} T_1 \\[1em] T_2 \\[1em]T_3 \\[1em] T_4\\[1em] T_5 \\[1em] T_6\\[1em] T_7 \\[1em] T_8\\[1em] T_9\\ \end{bmatrix} = \begin{bmatrix} -100 \\[1em] -20 \\[1em] -20 \\[1em] -80\\[1em] 0 \\[1em] 0\\[1em] -260 \\[1em] -180\\[1em] -180 \\ \end{bmatrix}\end{equation}$$ From this, your job using Gauss-elimination will be facilitated.

NOTE.- It could be that there are some lapses in the numbers written down. I think not, but maybe yes.