Explanation Given: There are six possible ways of offspring mate. State Mating 1 A A and A A 2 A A and A a 3 A A and a a 4 A a and A a 5 A a and a a 6 a a and a a where A is the dominant gene and a is the recessive gene. Now, verify that the transition matrix is: [ s 1 s 2 s 3 s 4 s 5 s 6 s 1 1 0 0 0 0 0 s 2 1 4 1 2 0 1 4 0 0 s 3 0 0 0 1 0 0 s 4 1 16 1 4 1 8 1 4 1 4 1 16 s 5 0 0 0 1 4 1 2 1 4 s 6 0 0 0 0 0 1 ] Explanation: Consider the state 1, that is, the parents carry genes A A and A A , then all the offspring carry the genes A A . Thus, the probability that pair of offspring in state 1 will be in state 1 is 1 and the probability that pairs of offspring are in any other state is 0. Write the matrix in the form: [ I m O R Q ] The fundamental matrix is: F = ( I − Q ) − 1 So, the matrix P : P = [ s 1 s 6 s 2 s 3 s 4 s 5 s 1 1 0 0 0 0 0 s 6 0 1 0 0 0 0 s 2 1 4 0 1 2 0 1 4 0 s 3 0 0 0 0 1 0 s 4 1 16 1 16 1 4 1 8 1 4 1 4 s 5 0 1 4 0 0 1 4 1 2 ] Thus, the matrix is: [ I 2 O R Q ] Therefore, the matrices R and Q are: R = [ 1 4 0 0 0 1 16 1 16 0 1 4 ] and Q = [ 1 2 0 1 4 0 0 0 1 0 1 4 1 8 1 4 1 4 0 0 1 4 1 2 ] Thus, the fundamental matrix is: F = ( I 4 − Q ) − 1 Then, F = ( I 4 − Q ) − 1 = ( [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] − [ 1 2 0 1 4 0 0 0 1 0 1 4 1 8 1 4 1 4 0 0 1 4 1 2 ] ) − 1 = [ 1 2 0 − 1 4 0 0 1 − 1 0 − 1 4 − 1 8 3 4 − 1 4 0 0 − 1 4 1 2 ] − 1 = A − 1 Now, compute the inverse of the matrix A as: [ 1 2 0 − 1 4 0 1 0 0 0 0 1 − 1 0 0 1 0 0 − 1 4 − 1 8 3 4 − 1 4 0 0 1 0 0 0 − 1 4 1 2 0 0 0 1 ] Multiply the row 1, 3, and 4 with 4, 8, and 4, respectively

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 10, Problem 63RE

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The fundamental matrix F and product matrix FR.

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Chapter 10, Problem 62RE

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The fundamental matrix F and product matrix F R .

Solution Summary: The author explains the fundamental matrix F and the product matrix, FR, and states that there are six possible ways of offspring mate.

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