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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 Boss (42.5k points)
edited by | 2.6k views

3 Answers

+29 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 Boss (42.5k points)
edited by
0
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.
+17

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 *

+2
will the answer change for homogeneous equations?
0

thank you .... it helped .. ur 'there exists'

+2
Statement 3) is " there exists a system which has a solution "  not  " there exists a solution "
0
ans should be option d ,    there may be cases where m=n  but those m equations be linearly dependent and results in no solutions so case iii will also be wrong
0
@sushmita in case of homogeneous equeations when m>n the max rank will be n and this boild down to r<=n hence solution exists in case of m<n max rank is m and r<n infinite solutions exist and when m=n r<=n hence solution exists in all the three cases
0 votes

I think here we required rank(A | B) and rank(A) to conclude clearly . Becoz;

FOR HOMOGENEOUS SYSTEM :-

1) INCONSISTENT ;- Not possible 

2) Unique solution :- Rank == No of variables

3) Many Solution :- Rank < No of variables.

FOR NON - HOMOGENEOUS SYSTEM :-

1) INCONSISTENT ;- when Rank(A|B) !=Rank (A) 

2) Unique solution :- when Rank(A|B) =Rank (A) and Rank == No of variables

3) Many Solution :- when Rank(A|B) =Rank (A) and Rank < No of variables.

So It has m linear eqn so maxm rank can be = "m"

and no of variables are = "n"

and to conclude anything Further we need to assume Rank (A|B)=rank(A).

After assuming this only Case (III) can be true.

Becoz m can not be greater than n (Becoz if so , then we can not conclude anything) , hence (II) is false.

When m<n , then all such system has Many solution, Hence (I) is false..

answered by Active (1.2k points)
–2 votes

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 (157 points)
+5

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 *



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