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Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If the 2x2 matrix P is the transition matrix for a regular Markov chain, then, at most, one of the entries of P is equal to 0. Choose the correct answer below. O A. This is false. The matrix P must be regular, which means that P can only contain positive entries. Since zero is not a positive number, there cannot be any entries that equal 0. O B. This is false. In order for P to be regular, the entries of P^k must be non-negative for some value of k. For k=1 the matrix 01 has non-negative entries and has two zero entries. Thus, it is a regular transition matrix with more than one entry equal to 0. OC. This is true. If there is more than one entry equal to 0, all powers of P will contain 0 entries. Hence, there is no power k for which pk contains all positive entries. That is, P will not satisfy the definition of a regular matrix if it has more than one 0. O D. This is true. If there is more than one entry equal to 0, then the number of entries equal to zero will increase as the power of P increases.

Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If the 2x2 matrix P is the transition matrix for a regular Markov chain, then, at most, one of the entries of P is equal to 0. Choose the correct answer below. O A. This is false. The matrix P must be regular, which means that P can only contain positive entries. Since zero is not a positive number, there cannot be any entries that equal 0. O B. This is false. In order for P to be regular, the entries of P^k must be non-negative for some value of k. For k=1 the matrix 01 has non-negative entries and has two zero entries. Thus, it is a regular transition matrix with more than one entry equal to 0. OC. This is true. If there is more than one entry equal to 0, all powers of P will contain 0 entries. Hence, there is no power k for which pk contains all positive entries. That is, P will not satisfy the definition of a regular matrix if it has more than one 0. O D. This is true. If there is more than one entry equal to 0, then the number of entries equal to zero will increase as the power of P increases.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If the 2x2 matrix P is the transition matrix for a regular Markov chain, then, at most, one of the entries of P is equal to 0. Choose the correct answer below. O A. This is false. The matrix P must be regular, which means that P can only contain positive entries. Since zero is not a positive number, there cannot be any entries that equal 0. O B. This is false. In order for P to be regular, the entries of P^k must be non-negative for some value of k. For k=1 the matrix [:] has non-negative entries and has two zero entries. Thus, it is a regular transition matrix with more than one entry equal to 0. 01 OC. This is true. If there is more than one entry equal to 0, all powers of P will contain 0 entries. Hence, there is no power k for which pk contains all positive entries. That is, P will not satisfy the definition of a regular matrix if it has more than one 0. O D. This is true. If there is more than one entry equal to 0, then the number of entries equal to zero will increase as the power of P increases.
Transcribed Image Text:Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If the 2x2 matrix P is the transition matrix for a regular Markov chain, then, at most, one of the entries of P is equal to 0. Choose the correct answer below. O A. This is false. The matrix P must be regular, which means that P can only contain positive entries. Since zero is not a positive number, there cannot be any entries that equal 0. O B. This is false. In order for P to be regular, the entries of P^k must be non-negative for some value of k. For k=1 the matrix [:] has non-negative entries and has two zero entries. Thus, it is a regular transition matrix with more than one entry equal to 0. 01 OC. This is true. If there is more than one entry equal to 0, all powers of P will contain 0 entries. Hence, there is no power k for which pk contains all positive entries. That is, P will not satisfy the definition of a regular matrix if it has more than one 0. O D. This is true. If there is more than one entry equal to 0, then the number of entries equal to zero will increase as the power of P increases.
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