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

Question

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.

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2 weeks 2021-09-15T07:28:26+00:00 1 Answer 0

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