GATE CSE 1996 | Question: 1.7

Question

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?

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

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

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

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

Comments

Corrected.

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Ans:C

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A) True

B) The homogeneous (with all constant terms equal to zero) underdetermined linear system always has non-trivial solutions.

https://en.wikipedia.org/wiki/Underdetermined_system

C) Can't claim anything about the nature of solution.

D) True

$A_{n*n}X_{n*1}$=0 & |A|$\neq$ 0 & $\rho (A)=n$, then trivial solution

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In option b they are talking about homogenous system of equation but how it can be infinite no of solution when nothing is given about rank .solution is infinite when rank<no of unknown.no such thing is given in option so how we can decide it is correct??

Answer 1

A)the system has a solution if and only if ,both A and augmented matrix[AB] have the same rank because [0 0 0 0 0 nonzero] does not exist .it is true.

B)it is true because if m<n there is possibility free variable exist , then there is infinite many solutions exist.

c)it is false because if m==n it does not mean that there exist a unique solution soltion depend on rank if rank[A]=rank[AB] then only unque solution exist.

d)it is true because rank=n and does not exist any free variable

Answer 2

(A) The system has a solution if and only if, both |A| and the augmented matrix |AB| have the same rank True

(B) If m<n and b is the zero vector, then the system has infinitely many solutions. True
 

(C) If m=n and b is a non-zero vector, then the system has a unique solution. False 

(D) The system will have only a trivial solution when m=n, b is the zero vector and rank(A)=n. True
 

Answer 3

First, they asked about False option

Option-A :   True  

When the rank[A] = rank[A,B] then only the system of linear equations has the solution  

If rank[A] is not equal to rank[A,B] then system of linear has No solution

Option-B: True

They have mentioned that b is a zero vector. So [000…0|b] this case is not possible. I.e No solution case is not possible

When m<n then at least one column will be free . So if there is at least one free column then the system of linear equation has infinite solutions

Option-C: False

Given b is a zero vector so [000…0|b] this case is not possible.i.e, No solution case is not possible 

They mention m=n but they did not mention the rank.

If the rank[A|b] = m then only the system has unique solution.

If the rank[A|b] < m then the system can have infinite solution Because there will be a free variable

P   0   0

-   P   0

-   -   0

0   0   0

Option-D: True

Given m=n and b is a zero vector so the equation is AX=0.

Now they mentioned that rank[A] = n. That is every column is linearly independent.

Important point here is all the columns are linearly independent so there will be no non-zero vector X  which makes AX=0. The vector must be zero vector (trivial) which results AX=0