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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.
edited | 3.2k views

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 (43.4k points)
edited
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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.
+25

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

+3
Statement 3) is " there exists a system which has a solution "  not  " there exists a solution "
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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
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@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
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for linear equations

a+b+c=2

a+b+c=3

a+b+c=5

system is inconsistent

so how option C is correct

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 (1.3k points)
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i  think homogeneous systems always have at least one solution, namely, the case where all unknowns are equal to zero.  i.e.trivial solution to the homogeneous system, so whether m> n or m<n or m=n homogenous will have always a solution and can never be inconsistent.

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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