Let P be an n × n stochastic matrix. The following

QUESTION:

Let P be an n X n stochastic matrix. The following argument shows that the equation Px = x has a nontrivial solution. (In fact, a steady-state solution exists with nonnegative entries. A proof is given in some advanced texts.) Justify each assertion below. (Mention a theorem when appropriate.)

a. If all the other rows of P – I are added to the bottom row, the result is a row of zeros.

b. The rows of P – I are linearly dependent.

c. The dimension of the row space of P – I is less than n.

d. P – I has a nontrivial null space.