please answer step by step with words or math explains.

Please answer the following questions, justifying your answers: a. Consider a market represented by an mn payoff matrix A and an n1 price vector S. If A is not invertible, how many linearly independent assets are there? b. Consider a market in which one of the state price vectors is =[101]. Is it true that in this market there is definitely arbitrage? c. Consider the following statement: If there exists a state price vector with some non-positive components, then there is arbitrage. Is this statement true? d. The annual log true return of a stock is i.i.d. normally distributed with mean and variance 0.12 and 0.12, respectively. You want to write a 6-period binomial model to price a derivative that expires in 6 months and whose payoffs depend on the price of this stock. What will be the high and low per-period return for the stock (i.e., Ru and Rd in the notation used in class) Please answer the following questions, justifying your answers: a. Consider a market represented by an mn payoff matrix A and an n1 price vector S. If A is not invertible, how many linearly independent assets are there? b. Consider a market in which one of the state price vectors is =[101]. Is it true that in this market there is definitely arbitrage? c. Consider the following statement: If there exists a state price vector with some non-positive components, then there is arbitrage. Is this statement true? d. The annual log true return of a stock is i.i.d. normally distributed with mean and variance 0.12 and 0.12, respectively. You want to write a 6-period binomial model to price a derivative that expires in 6 months and whose payoffs depend on the price of this stock. What will be the high and low per-period return for the stock (i.e., Ru and Rd in the notation used in class)