Linear system solutions Assume we have linear system in form
$$
Ax=b
$$
Then I know that the system has solution when $\text{rank} A =\text{rank} (A|b) $.
Can you tell me more properties like this one which helps us to know whether the system has any solutions or maybe 6 has only one solution using the determinant of matrix and other characteristics of it? 
 A: Let $A$ be an $m\times n$ matrix.


*

*If $\operatorname{rank}(A)=m$, there always exists a solution.

*If $\operatorname{rank}(A)=n$, there might be solutions or not, but if, then they are unique.

*Consequently, if $\operatorname{rank}=m=n$, there always exists a unique solution. One may rephrase this to saying that $\det(A)\neq 0$.

A: The determinant is always defined by the square matrix. Consequently, if a square matrix $A$ has non-zero determinant, this will lead to the invertibility and thus $x=A^{-1}b$ is the unique solution.
However, you cannot judge the existence and uniqueness of the linear equations by using the determinant if $A$ is not a square matrix. Simply consider like follows. Assume $A$ is a $m$-row, $n$-column matrix. If $A$ is full row rank which means that $A$ has $m$ linearly independent columns which can span the entire $m$ space (refer as range $R$). So that for any vector $b$ belongs to $R$, the solution always exists. 
On the other hand, if $rank(A)=n$, all the columns in $A$ are linearly independent and $x=0$ would be the only solution of $Ax=0$. Notice that the full column rank condition does not give any information on the range space. The only conclusion can be made is that, the solution is unique if it has solution.
