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Practice Set B: Calculus, Graphics, and Linear Algebra 89

appears in the corresponding column of the diagonal matrix R. This says
exactly that AU = UR.
(a) Verify the equality AU = UR for eachof the coefficient matrices in
Problem 10.
(b) In fact, rank(A) = rank(U), so when A is nonsingular, then

U −1 AU = R.

Thus if two diagonalizable matrices A and B have the same set of
eigenvectors, then the fact that diagonal matrices commute implies
the same for A and B. Verify these facts for the two matrices

   
1 0 2 52 −8
A =  −1 0 4  , B =  36 −10  ;
−1 −15 33 −7

that is, show that the matrices of eigenvectors are the “same” —
that is, the columns are the same up to scalar multiples — and
verify that AB = BA.
13. This problem, having to do with genetic inheritance, is based on Chapter
12 in Applications of Linear Algebra, 3rd ed., by C. Rorres and H. Anton,
John Wiley & Sons, 1984. In a typical inheritance model, a trait in the off-
spring is determined by the passing of a genotype from the parents, where
there are two independent possibilities from each parent, say A and a,
and eachis equally likely. (A is the dominant gene, and a is recessive.)
Then we have the following table of probabilities of the possible geno-
types for the offspring for all possible combinations of the genotypes of the
parents:

Genotype of Parents
AA-AA AA-Aa AA-aa Aa-Aa Aa-aa aa-aa
Genotype AA 1 1/2 0 1/4 0 0
of Aa 0 1/2 1 1/2 1/2 0
Offspring aa 0 0 0 1/4 1/2 1

Now suppose one has a population in which mating only occurs with
one’s identical genotype. (That’s not far-fetched if we are considering con-
trolled plant or vegetable populations.) Next suppose that x 0 , y 0 , and z 0
denote the percentage of the population with genotype AA, Aa, and aa
respectively at the outset of observation. We then denote by x n, y n, and
z n the percentages in the nthgeneration. We are interested in knowing