search
Log In
14 votes
1.8k views

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

  1. no solution
  2. a unique solution
  3. more than one but a finite number of solutions
  4. an infinite number of solutions
in Linear Algebra 1.8k views

2 Answers

21 votes
 
Best answer
rank of matrix $=$ rank of augmented matrix $=$ no of unknown $=$ $3$
so unique solution..

Correct Answer: $B$

edited by
0
Can case C arise? If Yes, how shall we determine?
0

when rank of matrix = rank of augmented matrixno of unknown 

3
then it is infinite solutions. r < n, that is option D. I'm asking about option C
1
i think more than one but a finite number of solutions will never arise

as we have only 3 cases r=n,r<n and r>n
2
yes c option case can never arise
0

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

0
How can we find determinant of augmented matrix ?
7 votes

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

1
in matrix $[A]_{3\times3},$ if $|A|_{3\times3}\neq0$ then rank should be $3$
1
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
Answer:

Related questions

37 votes
5 answers
1
7.6k views
Let $c_{1}.....c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}$ = 0 where $a_{i}$ are column vectors in $R^{n}$. Consider the set of linear equations $Ax = b$ where $A=\left [ a_{1}.....a_{n} \right ]$ ... set of equations has a unique solution at $x=J_{n}$ where $J_{n}$ denotes a $n$-dimensional vector of all 1. no solution infinitely many solutions finitely many solutions
asked Feb 14, 2017 in Linear Algebra Arjun 7.6k views
37 votes
4 answers
2
5.5k views
Consider the system, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the following is CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
asked Feb 12, 2016 in Linear Algebra Akash Kanase 5.5k views
22 votes
5 answers
3
3.8k views
If the following system has non-trivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0$, then which one of the following options is TRUE? $p - q + r = 0 \text{ or } p = q = -r$ $p + q - r = 0 \text{ or } p = -q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p - q + r = 0 \text{ or } p = -q = -r$
asked Feb 15, 2015 in Linear Algebra jothee 3.8k views
27 votes
3 answers
4
3.9k views
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false? The system has a solution if and only if, both $A$ ... system has a unique solution. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.
asked Oct 9, 2014 in Linear Algebra Kathleen 3.9k views
...