Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form where a and B are constants between 0 and 1. (There are two linearly 1- в. independent steady state vectors if Q = B = 0. Otherwise, there is only one.)

Determine whether or not the following sets S of 2 x 2 matrices are linearly independent. 25 -4 {(²5) (22)} 0 Select an Answer Select an Answer Select an Answer Select an Answer 1. S ✓2. S 3. S 4. S = = " = 25 {(²5). ( 22 )} -4 -10 -12 0 25 -4 -22 17 -31 -{(² {(5)). (1 3). (1 3) (5 2). (772)) ¹)} " -2 0 9 10 -10 -2 ㅠ e² = {(²9)· (42) (43)} 2 3 >

In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that will add a fourth counter. Counters 1,2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. (a) The steady-state probabilities, for all . (b) The probability that a fourth counter will be needed. (c) The average number of idle counters.

If A is an n×n matrix having linearly independent eigenvectors v1,v2, . . . ,vn, then the n×n matrix P= [v1v2. . .vn] is invertible.

If A is a 2 x 2 skew symmetric matrix, must det(A) = 0?