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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 $\text{rank}(A) =n$.

7 Answers

1 votes
1 votes
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
0 votes
0 votes
(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)=nTrue
 
0 votes
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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

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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

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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

Answer:

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