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


asked in Linear Algebra by Veteran (66.2k points) 1154 2199 2523 | 605 views
All are true, right?
yes, all should be right here.

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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 (2.3k points) 6 14 32
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If rank of A = rank of AB = n

then the solution would be unique

What about option (d) given m = n and if Rank(A) is n this means it would have n linear independent equations and n unknowns so it would have unique solution and trivial solution both but statement says "The system will have only a trivial solution". Please someone explain if i am wrong

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