Question

Linear Algebra

The question is actually on the third picture, i’m not sure if the information on picture 1 and 2 is supposed to help with the problem.

Please complete Q9. You may need information from Q7.

Please clearly label a b. Total 2 questions. Please complete Q 9.

Some problems in genetics can be solved using Markov chains. For example, if we know the genotype distribution of the present generation, call it xo, we can use the transition matrix A of the population to find the genotype distributio n x, of future generations. Let X, be the initial genotype distribution of a population. Assume that the transition matrix for the genotype of the given population is known and call it A. One year later, the genotype of the population, X,, is equal to the product of the transition matrix A the initial matrix Xo Two years later, the genotype of the population can be computed similarly:

Answer 1

9. Each flower is fertilised by a flower of its own genotype, therefore, there will be three cases:

First, consider the crossing of AA with AA, the offspring will be AA. The probabilities of an offspring to be AA, Aa and aa will be 1,0,0.

Consider the crossing of Aa with Aa, the offspring will be Aa. The probabilities of an offspring to be AA, Aa and aa will be 0,1,0.

Consider the crossing of aa with aa, the offspring will be aa. The probabilities of an offspring to be AA, Aa and aa will be 0,0,1.

It can be viewed as the following table:

Parents Genotype			Offspring Genotype
AA - AA	Aa-Aa	aa-aa
1	0	0	AA
0	1	0	Aa
0	0	1	aa

The transition matrix A:

001 010 100

Part B)

Genotypes in the 5th generation (or after 4 generations) will be:

$A^{4}x_{0}$ = $\begin{bmatrix} a_{0}\\ b_{0}\\ c_{0} \end{bmatrix}$