Example 1

Which one of the following matrices are productive?

a)

$$A= \left[ \begin{array}{rrr} 0.6 & .02 & 0.1\\ 0.3 & 0.2 & 0.4\\ 0.2 & 0.4 & 0. 3 \\ \end{array} \right]$$

b)

$$B= \left[ \begin{array}{rr} 0.6 & 0.5 \\ 0.1 & 0.5 \\ \end{array} \right]$$

solution

a) The matrix $A$ is productive, because each of its row adds to a number less than 1.

b) Matrix $B$ is productive because you can find a vector $x = (2, 1)$ such that $x > Bx$.