Chapter 4.4, Problem 72E

Heredity. Gregor Mendel (1822-1884) made discoveries that revolutionized the science of genetics. In one experiment he crossed dihybrid yellow round peas (yellow and round are dominant characteristics; the peas also contained genes for the recessive characteristics green and wrinkled) and obtained peas of the types indicated in the matrix:

$\begin{array}{l}\begin{array}{cc}\text{Round}& \text{Wrinkled}\end{array}\\ \begin{array}{c}\text{Yellow}\\ \text{Green}\end{array}\left[\begin{array}{cc}\text{315}& \text{ 101}\\ \text{108}& \text{ 32}\end{array}\right]=M\end{array}$

Suppose he carried out a second experiment of the same type and obtained peas of the types indicated in this matrix:

$\begin{array}{l}\begin{array}{cc}\text{Round}& \text{Wrinkled}\end{array}\\ \begin{array}{c}\text{Yellow}\\ \text{Green}\end{array}\left[\begin{array}{cc}\text{370}& \text{ 128}\\ \text{110}& \text{ 36}\end{array}\right]=N\end{array}$

If the results of the two experiments are combined, discuss matrix multiplication methods that can be used to find the following quantities. State the matrices that can be used and perform the necessary operations.

(A) The total number of peas in each category.

(B) The total number of peas in all four categories.

(C) The percentage of peas in each category