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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.

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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 55E
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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.
Transcribed Image Text: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.
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