GATE CSE
First time here? Checkout the FAQ!
x
+8 votes
1.3k views

Consider the system, each consisting of $m$ linear equations in $n$ variables.

  1. If $m < n$, then all such systems have a solution.
  2. If $m > n$, then none of these systems has a solution.
  3. If $m = n$, then there exists a system which has a solution.

Which one of the following is CORRECT?

  1. $I, II$ and $III$ are true.
  2. Only $II$ and $III$ are true.
  3. Only $III$ is true.
  4. None of them is true.

 

 

asked in Linear Algebra by Veteran (41.5k points)  
edited by | 1.3k views

2 Answers

+12 votes
Best answer
Correct answer => C)

why ?

I) This is false. Consider a system with m < n, which are incosistent like

a+b+c = 2

a+b+c = 3

Here m < n but no solution because of inconsistency !

II) m > n but no solution for none of system => What if this system of equations have 2 equations which are dependent ?

ex => a+b = 2

2a + 2b = 4

a-b = 0

Then a = 1, b= 1 is solutions . II) Is false.

III) this is true, M = 2, N = 2

a+b = 2

a-b = 0

Then m= 1, n= 1 Now there exists system which has solution . III) is correct. Answer is C !
answered by Veteran (41.5k points)  
selected by
but for the case of two parallel lines example y=x+5 and y=x+6 for these equations no solution so c should also be false.

You need to read the statement III) Clearly. What you are deducing is incorrect ! If m = n , the there exists a system which has a solution

* there exists *

Counter example is used for disproving, for all, not * there exists *

will the answer change for homogeneous equations?
–1 vote

Answer D:

I & II are false already but similarly for III also 

 for the case of two parallel lines example y=x+5 and y=x+6 for these equations no solution so c should also be false.

Hence D is correct ans.

answered by (163 points)  

You need to read the statement III) Clearly. What you are deducing is incorrect ! If m = n , the there exists a system which has a solution

* there exists *

Counter example is used for disproving, for all, not * there exists *



Top Users May 2017
  1. akash.dinkar12

    3152 Points

  2. pawan kumarln

    1616 Points

  3. sh!va

    1580 Points

  4. Arjun

    1326 Points

  5. Devshree Dubey

    1230 Points

  6. Angkit

    1028 Points

  7. Debashish Deka

    1012 Points

  8. Bikram

    972 Points

  9. LeenSharma

    810 Points

  10. srestha

    662 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. pawan kumarln

    242 Points

  2. Ahwan

    138 Points

  3. joshi_nitish

    112 Points

  4. jjayantamahata

    104 Points

  5. Aditya GN

    63 Points


22,725 questions
29,056 answers
65,052 comments
27,566 users