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Chapter 2: Q68E (page 86)

Question: Show that if A and B are nxn transition matrices, then AB will be a transition matrix as well. Hint: Use Exercise 67b.

Short Answer

Expert verified

Answer:

Hence, C is a transition matrix.

Step by step solution

01

Transition matrix 

For any transition matrix, the sum of the entries of the columns is equal to 1.

02

Proof

Let there be two nxn matrices A and B with non-negative elements. Then from exercise 67(b), we have:

A=e鈫�e鈫�B=e鈫�

Also, let C = AB . Then, again from 67(b), we get:

e鈫�C=e鈫�

In order to prove this, let C be the transition matrix. Then,

e鈫�C=e鈫�AB=e鈫�B=e鈫�

Hence, C is a transition matrix.

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Most popular questions from this chapter

If A is any invertiblen脳n matrix, then A commutes with A-1.

There exists a regular transition matrix A of size 3 脳 3 such that A2=A.

TRUE OR FALSE?

Matrix(111101110) is invertible.

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?24 f(x)=x3-x.

TRUE OR FALSE?

There exists a real number k such that the matrix[k-1-2-4k-3] fails to be invertible.
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