5. Stochastic and Stochastic n-by-n matrices Recall that an n × n matrix A is said to be stochastic if the following conditions are satisfied (a) Entries of A are non negative, that is ai, j ≥ 0 for all 1 ≤ i ≤ n and all 1 ≤ j ≤ n. (b) Each column of A sums to 1, that is Pn i=1 ai, j = 1, for all 1 ≤ j ≤ n. Let S and M be arbitrary stochastic n-by-n matrices. n matrices Recall that an n × n matrix A is said to be stochastic if the following conditions are satisfied

Which of the following tables represents exponential functions?

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?

How to simply and write the answer n a whole number