How to determine whether System has unique solution for $6 \times 3$ matrix?

Question

I am trying to solve a question where it is asked

Whether solution of the system is unique and if yes how. Details of the system are as given below :-

$1.$ System $AX = B$ is consistent.

$2.$ $A$ is a $6 \times 3$ matrix.

$ 3.$ Number of linearly independent rows in $A$ is $3$.

As linearly independent rows are $3$, so rank of matrix $A$ is $3$. As per my understanding if rank of matrix $=$ no. of unknowns, than system has unique solution. But in $6 \times 3$ matrix there can be $6$ unknowns, so is it possible that system can be unique, if yes how?

Answer 1

Theorem: Suppose that an $m×n$ system of linear equations is consistent and let $r$ be the rank of the system. If $n=r$, then the system has a unique solution. If $n>r$, then the system has infinitely many solutions.

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Since number of linearly independent rows in $A=(a_{ij})_{6 \times 3}$ is $3$, so rank of $A$ is $3$.

For your case $m=6, n=3, \text{and}, r=3$

Here $n=r$ and hence the system has unique solution.

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For further assistance you can visit the following link:

https://yutsumura.com/summary-possibilities-for-the-solution-set-of-a-system-of-linear-equations/