The solution to the steady-state heat flow problem for the plate in the figure is approximated by the solution to the equation Ax = b, where b = (5, 15, 0, 10, 0, 10, 20, 30) and

The missing entries in A are zeros. The nonzero entries of A lie within a band along the main diagonal. Such band matrices occur in a variety of applications and often are extremely large (with thousands of rows and columns but relatively narrow bands).

a. Use the method of Example 2 to construct an LU factorization of A, and note that both factors are band matrices (with two nonzero diagonals below or above the main diagonal). Compute LU - A to check your work.

b. Use the LU factorization to solve Ax = b.

c. Obtain A^-¹ and note that A^-¹ is a dense matrix with no
band structure. When A is large, L and U can be stored in
much less space than A^-¹. This fact is another reason for
preferring the LU factorization of A to A^-¹ itself.

Data from in Example 2 

A = 4 -1 -1 -1 -1 -1 in 4 0 -1 0 4 -1 4 0 -1 in -1 -1 0 0 1 -1 2 4 0-1 4 -1 -1 4 0-1 0 4 -1 -1 -1 4 -1 -1 0 0 3 5 7 6 8 10 10 10 10 20 20