...Continued
Question 3
A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique solution.
A - True, not 100% sure why though. B - False as a homogeneous system when m
A - True B - True as I have Verified by E x E^-1 = I
Thanks in advance for all your time and effort!
I'll answer only the question which's you're not sure $100\%$:-)
A $m\times n$ matrix $A$ is a representation of a linear transformation $$f\colon\Bbb R^n\rightarrow \Bbb R^m,\; x\mapsto Ax$$ and if $m<n$ then by rank-nullity theorem $f$ can't be injective and it could be non surjective so if $b\not\in\operatorname{im} f$ then the equation $Ax=b$ hasn't a solution but if $b\in\operatorname{im} f$ then the equation $Ax=b$ has infinity solutions.