Question

Discuss the validity of the following statement. If the statement is always​ true, explain why. If​ not, give a counterexample. If the 2 times 2 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. A. 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 Start 2 By 2 Table 1st Row 1st Column 0 2nd Column 1 2nd Row 1st Column 0 2nd Column 1 EndTable has​ non-negative entries and has two zero entries.​ Thus, it is a regular transition matrix with more than one entry equal to 0. B. 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. C. 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 Upper P Superscript k contains all positive entries. That​ is, P will not satisfy the definition of a regular matrix if it has more than one 0. D. 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.

Answer 1

C. 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 Upper P Superscript k contains all positive entries. That​ is, P will not satisfy the definition of a regular matrix if it has more than one 0

Explanation:

The correct option is C as it represents that by considering a matrix P that involves more than one zero and at the same time the powers for all P has received minimum one zero or it included at least one zero

Therefore the statement C verified and hence it is to be considered to be valid

Hence, all the other statements are incorrect