Secular Equations and Eigenvalues

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational 

Equation (3-106) is a simple first-order equation, whose solution is 

This result shows that we can write particular solutions to Eqs. (3-104) and (3-105) as 

These equations are satisfied if 7a = 0, 7z = 0, but this trivial solution is of no interest. To ensure a nontrivial solution, it is sufficient to require that the determinant of the coefficients of 7a and 7z be equal to zero, namely. 

---