Chapter 3.P1, Problem 1P

Assume that all the rate constants in $x_1{}^{\prime }=-\left(k_{01}+k_{21}\right)x_1+k_{12}{x}_2$, $x_2{}^{\prime }=k_{21}{x}_1-k_{12}{x}_2$ are positive.

(a) Show that the eigenvalues of the matrix $K=\left(\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right)=\left(\begin{array}{cc}-k_{01}-k_{21}& k_{12}\\ k_{21}& -k_{12}\end{array}\right)$ are real, distinct, and negative.

Hint: Show that the discriminant of the characteristic polynomial of $K$ is positive.

(b) If ${\lambda }_1$ and ${\lambda }_2$ are the eigenvalues of $K$, show that ${\lambda }_1+{\lambda }_2=-\left(k_{01}+k_{12}+k_{21}\right)$ and ${\lambda }_1{\lambda }_2=k_{12}{k}_{01}$.