##### Example4.5.5

The matrix $$A = \left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right]$$ is not positive. We can see this because some of the entries of $$A$$ are zero and therefore not positive. In addition, we see that $$A^2 = I\text{,}$$ $$A^3 = A$$ and so forth. Therefore, every power of $$A$$ also has some zero entries, which means that $$A$$ is not positive.

The matrix $$B = \left[\begin{array}{rr} 0.4 \amp 0.3 \\ 0.6 \amp 0.7 \\ \end{array}\right]$$ is positive because every entry of $$B$$ is positive.

Also, the matrix $$C = \left[\begin{array}{rr} 0 \amp 0.5 \\ 1 \amp 0.5 \\ \end{array}\right]$$ clearly has a zero entry. However, $$C^2 = \left[\begin{array}{rr} 0.5 \amp 0.25 \\ 0.5 \amp 0.75 \\ \end{array}\right] \text{,}$$ which has all positive entries. Therefore, we see that $$C$$ is a positive matrix.

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