Question

If AA and BB are stochastic n x n matrices, then ABTABT is also stochastic.

a. true
b. false

Answer 1

The statement is false because the row sums of the product ABTABT do not necessarily equal 1, which is a requirement for a matrix to be considered stochastic. Option b

Step-by-step explanation:

The question asks whether the product of two stochastic matrices AA and BB followed by their transpose ABTABT results in another stochastic matrix. First, we need to clarify that a stochastic matrix is a square matrix used to describe the transitions of a Markov chain, where each row sums to 1 and all entries are non-negative.

Now, by matrix multiplication rules, the product of two stochastic matrices is not guaranteed to be stochastic because the row sums might not be 1. However, if we take the transpose of a stochastic matrix, we end up with a matrix where each column sums to 1, not each row.

Therefore, the product ABTABT is not stochastic since it involves a product with a matrix that has column sums of 1, which does not adhere to the definition of a stochastic matrix having row sums of 1. Hence, the statement is false. Option b