Does there exist a matrix $A$ with vectors $p, q, r$ such that $Ax = p$ has no solution, $Ax = q$ has exactly one solution, and $Ax = r$ has infinitely many solutions? Justify your answer. If such a matrix does not exist, are there matrices for which two out of the three cases can hold? What are they?
I believe that there is such a matrix $A$ because if $Ax = q$ and $Ax = r$ has at least one solution, then that means that $q$ and $r$ are in the span of the vectors in the matrix $A$, which implies that $q$ and $r$ are multiples of the vectors in the matrix.
This also implies that the columns of $A$ are linearly dependent and therefore NOT invertible.
Do you think this is a proper justification? did i make any errors somewhere?