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

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?

• @user10354138 So you are saying that system given is is not consistent? – nbalodi Jun 2 at 13:05

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.

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.

For further assistance you can visit the following link:

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

• Thanks @nmasanta Link shared is quite useful to clear my doubts. – nbalodi Jun 2 at 14:11
• You are most welcome @nbalodi – nmasanta Jun 2 at 14:15