###### Example 4.5.5

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.