Linear Algebra True / False with justifications. Which of the following are correct or incorrect?

A) The determinant function $\det:M_n\rightarrow\Bbb R$ is a linear transformation.
False - determinants tell us areas or volumes.
B) If $A$ is an $n \times n$ and the matrix $B$ is obtained from $A$ by switching two rows, then $\det B=(-1)^n\det A$.
False  $\det -A=(-1)^n\det A$  However, this is true if $B=-A$.
C) If $A$ is an $n \times k$ matrix then $\operatorname{rank} A+\operatorname{null}(A)=n$.
False - $\operatorname{rank} A+\dim(\operatorname{null}(A))=n$.
D) The subset $S$ of $P_2$ consisting of all polynomial of the form $p(x)=x^2+14$ is a subspace of $P_2$.
False - Doesn't contain the zero vector.
E) If $A$ is an $n \times n$ matrix, then $\det A^T=\det A$.
True

The one major problem I see is in your reasoning for $(B)$. 
It is indeed false, but it is false because whenever you take $A$ and "swap one row with another" to get $B$, then $\det B = - \det A$, and so for even $n,\;$ the option $\;B = (-1)^n \det A = \det A \neq -\det A$, is indeed false.

Remarks:
I suspect that your justification for (C), which is correct, is likely what was intended by the statement, but your justification makes your answer clear, with respect to why you are concluding that, as literally stated, it is "false".
Suggestion: you need more justification for $(A)$. "False" is correct, but what does the fact that the determinant evaluates to a scalar (which in 2D, its absolute value represents area, and in 3D volume, but what about n-dimensions?) say about the $\det$ function being a linear transformation? You need to provide a justification which explains why it is not a linear transformation.
