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

For the record:
(A) holds if and only if $n=1$. (When $n\neq1$, $\det(2I)=2^n\neq2=2\det(I)$ shows it is wrong.)
(B) holds if and only if $n$ is odd or $\det(A)=0$.
(C) is a nonsensical statement as printed, since $\operatorname{null}(A)$ is a subspace, which cannot be added to a number. However taking the dimension of that subspace instead does not make the statement valid, since rank-nullity states $\operatorname{rank} A+\dim(\operatorname{null}(A))=k$ in this case (the dimension of the space of vectors $A$ can be applied to); the statement would only become true for $n=k$.
In the remaining cases the answer by OP is correct.
