11 I believe your proof is correct. Note that the best way of proving that $\det (A)=\det (A^t)$ depends very much on the definition of the determinant you are using. My personal favorite way of proving it is by giving a definition of the determinant such that $\det (A)=\det (A^t)$ is obviously true.
How do I prove that $\det A= \det A^T$? - Mathematics Stack Exchange
Given two square matrices $A$ and $B$, how do you show that $$\det (AB) = \det (A) \det (B)$$ where $\det (\cdot)$ is the determinant of the matrix?
linear algebra - How to show that $\det (AB) =\det (A) \det (B ...
It would be a good exercise to determine for which matrices, the identity $\det (A+b)=\det (A)+\det (B)$ holds. I think that it would work for rather few of them.
linear algebra - Does $\det (A + B) = \det (A) + \det (B)$ hold ...
In this case, obviously, $\det (A)=\det (A^*)$, but this is not generally true. You can decompose taking the conjugate transpose in two steps: first conjugate each entry, then transpose.
linear algebra - Describe $\det (A^*)$ in terms of $\det (A ...
Prove $$\det \left( e^A \right) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}^{n \times n}$.
linear algebra - How to prove $\det \left (e^A\right) = e ...
I hope I am not making any mistake but what the link says for this case is that determinant of sum, is sum of determinants of $2^n$ matrices which are constructed by choosing for each column i either ith column of A or ith column of B (all possible choices are $2^n$ if you think about it).