## Let A be a 3times3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. a. Does the equation Upper A B

Let A be a 3times3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. a. Does the equation 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.

## Answers ( )

Answer:the answer is in the explanation

Step-by-step explanation:For example, two pivot system

1 0 2 X

0 1 3 Y = 0

0 0 0 Z

X + 2z = 0

x = -2z

y + 3z = 0

y = -3z

free variables

z = z

x -2

y = -3

z = 1

(x, y, z) = (-2, -3, 1)

so the option D is correct

since A has 2 pivots

there is 1 free variable

so ax = 0