Let A be a 3times3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. a. Does the equati
on Upper A Bold x equals Bold 0 have a nontrivial solution? A. No. Since A has 2 pivots, there are no free variables. With no free variables, Upper A Bold x equals Bold 0 has only the trivial solution. B. Yes. Since A has 2 pivots, there is one free variable. The solution set of Upper A Bold x equals Bold 0 does not contain the trivial solution if there is at least one free variable. C. No. Since A has 2 pivots, there is one free variable. Since there is at least one free variable, Upper A Bold x equals Bold 0 has only the trivial solution. D. Yes. Since A has 2 pivots, there is one free variable. So Upper A Bold x equals Bold 0 has a nontrivial solution. b. Does the equation Upper A Bold x equals Bold b have at least one solution for every possible b? A. Yes. A has a free variable. So the free variable can equal any value such that there is at least one solution for every possible b. B. No. A has one free variable. To have at least one solution for every possible b, A cannot have any free variable. C. Yes. Since A has 2 pivots, there are no free variables. So there is at least one solution for every possible b. D. No. A has one free variable, so there will be no solution to the system for any possible b.
