Use the system in Problem 4 to find the all norms of the matrix A, the Eigenvalues, and the condition number. Is the system ill conditioned? Justify your answer. In all problems, rewrite the system in the form Ax=b.

 

P4: The following system of equations is designed to determine concentrations (the c’s in g/m3) in a series of coupled reactors as a function of the amount of mass input to each reactor (the right-hand sides in g/day):
15c1 - 3c2 - c3 = 4000
-3c1 + 18c2 - 6c3 = 1200 -4c1
- c2 + 12c3 = 2350

(a) Determine the matrix inverse.

(b) Use the inverse to determine the solution.

(c) Determine how much the rate of mass input to reactor 3 must be increased to induce a 10 g/m3 rise in the concentration of reactor 1.

(d) How much will the concentration in reactor 3 be reduced if the rate of mass input to reactors 1 and 2 is reduced by 500 and 250 g/day, respectively?