Exercise 1.4.29

State whether each of the following sets of data is possible for a system of equations. If possible, describe the solution set. That is, indicate whether there exists a unique solution, no solution or infinitely many solutions. Here, $A$ is the coefficient matrix, and $\left[A\mid B\right]$ denotes the augmented matrix of the system.

1. $A$ is a $5\times 6$-matrix, $\operatorname{rank}(A)=4$ and $\operatorname{rank}\left[A\mid B\right]=4$.

2. $A$ is a $3\times 4$-matrix, $\operatorname{rank}(A)=3$ and $\operatorname{rank}\left[A\mid B\right]=2$.

3. $A$ is a $4\times 2$-matrix, $\operatorname{rank}(A)=4$ and $\operatorname{rank}\left[A\mid B\right]=4$.

4. $A$ is a $5\times 5$-matrix, $\operatorname{rank}(A)=4$ and $\operatorname{rank}\left[A\mid B\right]=5$.

5. $A$ is a $4\times 2$-matrix, $\operatorname{rank}(A)=2$ and $\operatorname{rank}\left[A\mid B\right]=2$.