There is a rectangular plate whose edges are maintained at a certain temperature.

We are interested in finding the temperature at the interior points. Consider the

following diagram.

Approximations must be found for points $T_1$ to $T_9$; that is$,$ the temperature of the

intermediate points. Suppose that the temperature at an interior point is the

average of the temperatures of the four points surrounding it$.$ With this assumption, establish a system of equations, considering first the point

$T_1,$ then the point $T_2,$ etc. Rewrite the system so that all variables are on only one side of the equation.

Find the coefficient matrix and the augmented matrix. Observe the shape of the coefficient matrix $($diagonal$,$ band, symmetric, etc.$).$

Solve the system using the Gauss, Jacobi, and Gauss$-$Seidel methods. Make a table and compare your results.