in Linear Algebra
5,485 views
20 votes
20 votes

Consider the following set of equations

  • $x+2y=5$
  • $4x+8y=12$
  • $3x+6y+3z=15$

This set

  1. has unique solution
  2. has no solution
  3. has finite number of solutions
  4. has infinite number of solutions
in Linear Algebra
5.5k views

11 Comments

edited by
A|B matrix will be given as -

$\begin{bmatrix} 1 &2 &0 &|5 \\ 4& 8 &0 & |12 \\ 3& 6 & 3 & |15 \end{bmatrix}$

After the row-operations -   R3->R3-3R1  and

                                            R2->4R1-R2

We obtain the matrix -

  $\begin{bmatrix} 1 &2 &0 &|5 \\ 0 & 0& 0& |8\\ 0& 0 & 3 & |0 \end{bmatrix}$

Now since, $Rank(A)\neq Rank(A|B)$

Hence, this is an inconsistent solution, that is, no solution.
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@shweta how did you get 0000 oin the last row?
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After performing row operation R3 -> R3 - 3R1
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how 3 -0=0?
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My mistake it should be 3. I edited it.
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Formatting Error in PDF

There is an error in the formatting of this question in the soft copy of the GO PDF (screenshot attached for reference).

P.S: I am unable to add the screenshot. Please help me fix this.

 

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..

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The error is still there in GO hardcopy- 2019 

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With this print mistake, if we solve the question we get Option A) has unique solution. :P

Please let me know if I'm wrong. :)
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@rohith1001 Yes I too got the same option.

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Constant matrix also linearly dependent with same factor as coefficient matrixYes (Then Inconsistent System of Equation- No Solution) If →No (Then consistent System of Equation- Infinite Solutions)

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2 Answers

39 votes
39 votes
Best answer

There are no solutions.

If we multiply $1^{\text{st}}$ equation by $4,$ we get

  • $4x + 8y = 20$

But $2^{\text{nd}}$ equation says

  • $4x + 8y = 12$

Clearly, there can not be any pair of $(x,y),$ which satisfies both equations.

Correct Answer: B.

edited by

2 Comments

I am getting rank of augmented matrix as 3 as well as rank of A =3 so according to this condition there must be a unique solution , although it shouldn't exist but then how can it contradict ?
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Rank of A is not 3. Recheck your method.
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12 votes
12 votes

First transformed the matrix into echelon form and rest can seen from the pic.

Answer:

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