$m$ number of Rows
$n$ number of columns
true or false
A) If $n > m$, given any $b$ you can always solve $Ax=b$.
The answer: False. Counterexample: A is the zero matrix.
We have unknowns more than equations, so we can always solve $Ax=b$. Why the answer is false? And even if $A$ is the zero matrix we have the matrices x = zeros then $b$ zeros so there is always answer!! Can anyone explain why the answer is false? How can he proved false by $A$ is the zero matrix?
B) If $n < m$, the only solution of $Ax=0$ is $x=0.$ The answer: False. Counterexample: let $A$be the zero matrix.
I understand why it is false but I'm wondering, is there any special case makes this statement true?
Edit: I made a matrix $A= 2$ rows $\times 1$ column contains $(1,0)$ and assume Ax=0 as the statement, then the only solution is when x=0 because we can't use row echelon form any more. is what I did correct to make the statement true?