4. A population of cells has two phases of life: early and late. Every month, 40% of early-stage cells create a new early-stage cell and 60% of late-stage cells create a new early-stage cell. After a month, all early-stage cells survive to become late-stage cells, and 80% of late-stage cells survive another month. [E(t)] Let p(t) = (L(0)] be a population vector which represents the population of cells after t month. (a) Give a matrix equation which describes how the population changes from month t to month t+1. (b) Find the eigenvectors and eigenvalues of the transition matrix. (C) Diagonalize the transition matrix; i.e. write the transition matrix as the product of three matrices PDP-1, where D is a diagonal matrix. (d) If we start with just 100 early-stage cells (and no late-stage cells), then how many early-stage cells will we have in a year? (e) If we find that there are a total of 10,000 cells, how many cells can we expect after another month? 4. A population of cells has two phases of life: early and late. Every month, 40% of early-stage cells create a new early-stage cell and 60% of late-stage cells create a new early-stage cell. After a month, all early-stage cells survive to become late-stage cells, and 80% of late-stage cells survive another month. [E(t)] Let p(t) = (L(0)] be a population vector which represents the population of cells after t month. (a) Give a matrix equation which describes how the population changes from month t to month t+1. (b) Find the eigenvectors and eigenvalues of the transition matrix. (C) Diagonalize the transition matrix; i.e. write the transition matrix as the product of three matrices PDP-1, where D is a diagonal matrix. (d) If we start with just 100 early-stage cells (and no late-stage cells), then how many early-stage cells will we have in a year? (e) If we find that there are a total of 10,000 cells, how many cells can we expect after another month