Let $Ax = b$ be a system of linear equations where $A$ is a $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is a $n \times 1$ column vector of unknows. Which of the following is false?

The system has a solution if and only if, both $A$ and the augmented matrix $[Ab]$ have the same rank.

If $m < n$ and $b$ is the zero vector, then the system has infinitely many solutions.

If $m = n$ and $b$ is non-zero vector, then the system has a unique solution.

The system will have only a trivial solution when $m = n,$ $b$ is the zero vector and $rank (A) = n$.

"If m=n and b is non-zero vector, then the system has a unique solution" is FALSE actuallty.
"If m=n and b is non-zero vector AND rank of matrix is also m, then the system has a unique solution" is TRUE.