35.2. (This problem requires a knowledge of elementary linear algebra.) If A is a constant matrix, then Ñ(t + Ar) = AÑ(t) is a linear system of constant coefficient first order difference equations. The solution can be obtained by a method similar to that developed in Sec. 33. (d) A stable age distribution exists if the populations approach, as time increases, an age distribution independent of time. Show that if there exists a real positive eigenvalue (call it r₁) which is greater than the absolute value of the real part of all the other eigenvalues, then a stable age distribution exists. Show that a stable age distribution is C₁, where C, is an eigenvector corresponding to the eigenvalue r₁. Note that such a stable age distribution is independent of the initial age distribu- tion; it only depends on the birth and death rates of each age grouping! (e) If a stable age distribution exists, show that the total population grows like eat, where σ = In r₁. Thus show that the population exponentially grows if r₁> 1, exponentially decays if r₁ <1, and approaches a constant if r₁ = 1.