The band matrix A shown below can be used to estimate the unsteady conduction of heat in...

The band matrix A shown below can be used to estimate the unsteady conduction of heat in a rod when the temperatures at points p1,….,p5 on the rod change with time.

The constant C in the matrix depends on the physical nature of the rod, the distance between the points on the rod, and

the length of time between successive temperature measurements. Suppose that for k = 0, 1, 2,.. .. a vector in R⁵ lists the temperatures at time If the two ends of the rod arc maintained at 0°. then the temperature vectors satisfy the equation

a. Find the LU factorization of A when C =1.Arnatrix such as A with three nonzero diagonals is called a tridiagonal matrix. The L and U factors are bidiagonal matrices.

b. Suppose C= I and t₀= (10, 12, 12, 12, 10). Use the LU factorization of A to find the temperature distributions t₁, t₂, t₃, and t₄.