a linear-algebra problem in genetic inheritance

Suppose there is a pair of genes A and a. And there are three types of genotype, AA,*Aa* and aa.

Now, study the problem of genetic transition. That's, given an initial genotype fraction, say AA for 1/2, Aa for 1/2 and aa for 0 in an population.

After n generation, what will the fraction for each genotype be?

It is reasonable to construct a transition matrix.

And have the following equation.

Xn=A*Xn-1, where Xn-1 is the (n-1)th generation genotype fraction vector, Xn the similar meaning and A is the transition matrix mentioned above.

Now the problem is how to construct the transition matrix A.

I referenced a link on web, which follows the idea like the following:

For each genotype(AA,*Aa* and aa) to cross with AA( This is where I am confused with), the fraction or probability for getting each genotype is:

For AA to cross with AA

• 1 probability to get AA
• 0 probability to get Aa
• 0 probability to get aa

For Aa to cross with AA

• 1/2 probability to get AA
• 1/2 probability to get Aa
• 0 probability to get aa

For aa to cross with AA

• 0 probability to get AA
• 1 probability to get Aa
• 0 probability to get aa

And combining the three column vectors yields the transition matrix A, where is

1 1/2 0
0 1/2 1
0  0  0


So, why is it like this? The reference on the link is over. More details see here

And what I had in mind is like this:

Why consider the cross with AA (the dominant gene type,(in converse, the recessive gene type)) every time when constructing the transition matrix's columns. This is where I marked above.

Why not set Aa or aa as the basis?

For example, when setting Aa as the basis, instead of AA.

The transition matrix will be

1/2 1/2  0
1/4 1/2 1/4
0  1/2 1/2


And the result(the next generation's gene fraction) will be different.

In summary, my question is:

Is it necessary to set AA as the basis for each crossing when building the transition matrix?

PS: It may be more of a biological problem on selecting the crossing basis. I will try to seek help from that biology field.

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It is because, as the web site you are referring to explains, "Let us consider a series of experiments in which we keep crossing offspring with dominant animals only. Thus we keep crossing $AA$, $Aa$, and $aa$ with $AA$". In other words, the reason they chose $AA$ is simply a biological one.