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PLEASE ANSWER ALL QUESTIONS WITH CLEAR STEPS AND EXPLANATIONS

Question 3 (25 marks): The chemical concentrations \( C_{1} \) and \( C_{2} \) of two re-agents in a reaction satisfy the coupled differential equations \[ \begin{array}{l} \frac{d C_{1}}{d t}=-7 C_{1}+2 C_{2} \\ \frac{d C_{2}}{d t}=-12 C_{1}+3 C_{2} \end{array} \] (a) Write the two differential equations as one matrix equation. (b) Find the eigenvalues and eigenvectors of the matrix in (a) [2 marks] (c) Write down the general solution of the differential equations. and \( C_{2}(t) \) if \( C_{1}(0)=3 \) and \( C_{2}(0)=2 \) (d) A catalyst is added so that the two equations become [6 marks] \[ \begin{array}{l} \frac{d C_{1}}{d t}=-7 C_{1}+2 C_{2} \\ \frac{d C_{2}}{d t}=-12 C_{1}+2 C_{2} \end{array} \] Show that the solutions now oscillate with dissipation.