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

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    2021-09-15T07:29:30+00:00

    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

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45:7+7-4:2-5:5*4+35:2 =? ( )