GATE CSE
First time here? Checkout the FAQ!
x
+4 votes
312 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?

  1. The system has a solution if and only if, both $A$ and the augmented matrix $[Ab]$ have the same rank.

  2. If $m < n$ and $b$ is the zero vector, then the system has infinitely many solutions.

  3. If $m=n$ and $b$ is a non-zero vector, then the system has a unique solution.

  4. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.

 

asked in Linear Algebra by Veteran (58.2k points)   | 312 views
All are true, right?

1 Answer

+5 votes
Best answer

Ans would be C because it is a case of linear non-homogeneous equations so by having m = n, we can't say that it will have unique solution. Solution depends on rank of matrix A and matrix [ A B ].

If rank[ A ] = rank[ A B ], then it will have solution otherwise no solution

answered by Active (2.2k points)  
selected by
If rank of A = rank of AB = n

then the solution would be unique
Yeah

Related questions

Top Users Feb 2017
  1. Arjun

    5166 Points

  2. Bikram

    4204 Points

  3. Habibkhan

    3748 Points

  4. Aboveallplayer

    2986 Points

  5. sriv_shubham

    2298 Points

  6. Debashish Deka

    2234 Points

  7. Smriti012

    2142 Points

  8. Arnabi

    1998 Points

  9. mcjoshi

    1626 Points

  10. sh!va

    1552 Points

Monthly Topper: Rs. 500 gift card

20,815 questions
25,974 answers
59,606 comments
22,025 users