Suppose that A and B are arbitrary matrices. Is the quantity given always equal to
Recall that while multiplication is not commutative, addition is commutative. Hence . By applying the distributive properties, we can see that:
Observe equations and . We see that:
from the commutativity of addition. Hence, . They are equal.
Recommended textbook solutions
Construct the matrix diagram for each of the following models. Yellow perch is a species of freshwater fish. Suppose a population of interest lives to a maximum age of 4 years. Only 5% of age 1 individuals survive to age 2, 20% of age 2 individuals survive to age 3, and 75% of age 3 individuals survive to age 4. Both age 3 and age 4 individuals reproduce, producing 100 and 150 age 1 offspring, respectively. Using for the number of individuals of age i at time t, we have