Short Answer

If A is any transition matrix and B is any positive transition matrix, then AB must be a positive transition matrix.

The statement is false.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Explaining

Let A be a transition matrix and B a positive transition matrix. By definition, every column of A is a distribution vector and sums to one. Thus, A cannot contain any all-zero columns. However, A could contain an all-zero row. For example, the matrix $[\begin{matrix} 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ \frac{1}{2} & \frac{3}{2} & 1 \end{matrix}]$ is a transition matrix.

Step 2: Result

The (i,j)th entry of the matrix AB is the dot product of the ith row of A and the column of B. Since the row of A could be all zero, the dot product could be zero. Therefore, the matrix AB is not necessarily a positive transition matrix and the answer is False.

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Linear Algebra With Applications

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Short Answer

If A is any transition matrix and B is any positive transition matrix, then AB must be a positive transition matrix.

Step by Step Solution

Step 1: Explaining

Step 2: Result

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Linear Algebra With Applications

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Short Answer

If A is any transition matrix and B is any positive transition matrix, then AB must be a positive transition matrix.

Step by Step Solution

Step 1: Explaining

Step 2: Result

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