GATE CSE 1998 | Question: 1.2

Question

Consider the following set of equations

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

This set

• A. has unique solution
• B. has no solution
• C. has finite number of solutions
• D. has infinite number of solutions

Comments

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 matrix →Yes (Then Inconsistent System of Equation- No Solution) If →No (Then consistent System of Equation- Infinite Solutions)

Answer 1

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.

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.

Answer 2

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