GATE CSE 2005 | Question: 48

Question

Consider the following system of linear equations : $$2x_1 - x_2 + 3x_3 = 1$$ $$3x_1 + 2x_2 + 5x_3 = 2$$ $$-x_1+4x_2+x_3 = 3$$ The system of equations has

• A. no solution
• B. a unique solution
• C. more than one but a finite number of solutions
• D. an infinite number of solutions

Answer 1

rank of matrix $=$ rank of augmented matrix $=$ no of unknown $=$ $3$
so unique solution..

Correct Answer: $B$

Comments

@Angkit   rank(r)>n  this case will never arise

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How can we find determinant of augmented matrix ?

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@Jhaiyam then I must ask you how can you numbers of stars in the sky:) LOL

determinant is only defined for square matrix and augmented matrix is not a square matrix.

Answer 2

Determinant of matrix =14 which is non zero

If The determinant of the coefficient matrix is non zero then definitely the system of given equation has a unique solution 

 so option B

Comments

in matrix $[A]_{3\times3},$ if $|A|_{3\times3}\neq0$ then rank should be $3$

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if we get |A|=0 then we have to check for either infinite solⁿ or no solution ...so we have to go with our fundamental method..

Then i think finding determinant is not fruitful

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|A|=0 implies for a non-homogenous system  implies that the solution may be inconsistent or infinitely many.

Answer 3

                                                       (B) A unique solution .

 

Answer 4

Method-1:

2   3   -1

-1   2   5

3   5   1

In this matrix we can clearly see that three columns are Linearly independent and 3 L.I columns in R3 space

 

So It will fill the space and Ax=b 

 

So there will be unique solution

 

If the column space is filled and Ax=0 then there will be a trivial solution

Method-2: Converting into  echelon form

After converting into echelon form we obtain augmented matrix as

2   0   0

-1   7   0

3   1   32

1   1   46

We can clear see all the columns have pivot and there is no [00..00|b] form

 

So there will be unique solution

Method-3: By using rank

rank[A] =3 and rank[A|b]=3  and number of columns =3

Therefore unique solution is possible

Answer is option-B