Please just answer d) and e)The last time I posted it and due to the rules only a),b),c) were answered.Much apprciated! In this question, the real numbers in your answers should be given with three significant digits of accuracy after thedecimal point.The public health authorities of a small town have divided the population into three categories: covid-negative,covid-positive, and hospitalised. After performing regular, extensive tests, they have observed that in each successiveweek:• Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised.• Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to behospitalised.• Among those who are hospitalised, 60% recover and become negative, 30% are released from hospital butremain positive, and 10% remain hospitalised.(a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h representthe number of people in the population who are negative, positive, and hospitalised respectively, write down amatrix M for which [n', p',h'] = M[n; p; h], where [n',p', h'] represents the column vector of negative, positive,and hospitalised members of the population in the following week.(b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way thatd1 > A2 > A3. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen commandin Pari-GP.)(C) Write down eigenvectors V1, V2, V3 attached to A1, A, and A3 respectively.(d) Show that v1 can be normalised so the the sum of its coordinates is equal to 1, and, with this normalisation,show that, for any triple v = [n; p; h],= Pv1 + av2 + bv3 where P = n+p+h is the total population, and a and b are some real numbers (which you do not need tobother calculating.)(e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of peoplein the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of thepopulation. What are these percentages?

We are given the following information. The population has been divided into three categories:…

where P =n+p+h is the total population, and a and b are some real numbers (which you do not need to bother calculating.) (e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of people in the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of the population. What are these percentages?