$m \times n$ matrix where $m < n$ So I'm a long distance student and I need some help to bounce ideas off of other people who understand the work. Fellow students are few and far between. So while this is an assignment question, I just need help to check if my logic and understanding is correct
Question
Suppose $A$ is an $m \times n$ matrix where $m < n$: Which of the following statements is/are true?
A. The non-homogeneous system $Ax =b$ has at least one solution.
B. The homogeneous system $Ax = 0$ has a unique solution.
Answer A: I don't think A is correct. $Ax=b$ will only have at least one solution IF the linear system is consistent. In our study guide it said that if $m < n$ then infinitely many solutions exist. But I think this is a trick question. 
B: True, homogenous systems always has the trivial solution which is a unique solution.
A: Both are false. For example, consider $m = 3$ and $n = 4$.
For part A, consider:
$$
A = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix} \qquad\text{and}\qquad
b = \begin{bmatrix}
0 \\ 0 \\ 7
\end{bmatrix}
$$
Since the third row of $A$ is all zeroes, any linear combinations of its columns will give us a $0$ in the third entry (and not $7$).
For part B, use the same matrix $A$ from part A and observe that (besides the trivial solution), another solution to the homogeneous system is:
$$
x = \begin{bmatrix}
0 \\ 0 \\ 8 \\ 9
\end{bmatrix}
$$
A: Hints.


*

*A is false in general, as shown by the system
$$\begin{cases}
 x+y+z=0\\
 x+y+z=1
 \end{cases}$$

*B is definitely false, because you have more unknowns than equations.
