A city is split in three boroughs, denoted by M, B, and ). Assume the following. In 2020, 20% of the population lived in M, 35% in B, and 45% in Q. It has been observed that each year, the following moves occur. Out of the population of M, 60% remain, 10% move to B, and 30% to Q. Out of the population of B, 80% remain, 15% move to @, and 5% to M. Out of the population of Q, 70% remain, 20% move to B, and 10% to M. Denote by X the column vector that contains the proportion of the population living in each borough in 2020, in the order M, B, Q. Write down this vector (no need to explain). Denote by P a 3 x 3 matrix where the element P;; is the proportion of people moving from borough j to borough i (with the same order: borough 1 /2 / 3 is M/B/Q). Write this matrix P (no need to explain). Compute directly (without using matrices) the percentage of the population living in Q in 2021. . Denote by X; the column vector that contains the proportion of the population living in each borough in 2021. Write down this vector in terms of X, and P. Make sure to explain your reasoning. Finally, compute X, explicitly. . In general, denote by X, the column vector that contains the proportion of the population living in each borough in 2020 + k years. Write down this vector in terms of X, P, and k. Make sure to explain your reasoning. . A calculator allow to compute that 0.17 0.16 0.15 0.15 X2 ~ | 042 . X5 ~ | 0.45 y Xlg ~ | 0.46 5 XQO ~ | 0.46 0.41 0.39 0.39 0.39 What do vou observe? What does it mean in non technical terms