Consider a CSTR in which the following reactions occur: $AB,BC,CB.$ The reaction rates per unit volume of the three reactions are $r_1=k_1{C}_A,r_2=k_2{C}_B,$ and $r_3=k_3{C}_C$ respectively. The reactor has constant volumetric flow rate $q$ and pure component A feed at concentration $C_{Ai}.$ The volume of the reactor is denoted $V.$ Steady$-$state mass balances on the three reaction species yield the following algebraic equation system:

$\frac{q}{V}(C_{Ai}-C_A)-k_1{C}_A=0,-\frac{q}{V}{C}_B+k_1{C}_A-k_2{C}_B+k_3{C}_C=0,-\frac{q}{V}{C}_C+k_2{C}_B-k_3{C}_C=0$

$($a$)$ Given the parameter v$$lues $k_1=1,k_2=2,k_3=3,\frac{q}{V}=1,$ and $C_{Ai}=6,$ show that the linear algebra problem for determining $C_A,C_B,$ and $C_C$ can be posed as $Ax=b$ where:

$A=\left[\begin{array}{ccc}-2& 0& 0\\ 1& -3& 3\\ 0& 2& -4\end{array}\right],b=\left[\begin{array}{c}-6\\ 0\\ 0\end{array}\right]$

$($b$)$ Use Gaussian elimination to solve the linear algebraic system.