GATE CSE
First time here? Checkout the FAQ!
x
+6 votes
807 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 (40.7k points)  
edited by | 807 views

2 Answers

+9 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 (40.7k 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 *

–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 Jan 2017
  1. Debashish Deka

    8126 Points

  2. sudsho

    5042 Points

  3. Habibkhan

    4706 Points

  4. Vijay Thakur

    4458 Points

  5. Bikram

    4348 Points

  6. saurabh rai

    4212 Points

  7. Arjun

    4010 Points

  8. santhoshdevulapally

    3722 Points

  9. GateSet

    3292 Points

  10. Sushant Gokhale

    3286 Points

Monthly Topper: Rs. 500 gift card

19,122 questions
24,034 answers
52,724 comments
20,276 users