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Q32E

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Found in: Page 108

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# 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.

See the step by step solution

## 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 $\left[\begin{array}{ccc}0& 0& 0\\ 1}{2}& 1}{4}& 0\\ 1}{2}& 3}{2}& 1\end{array}\right]$ 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.