What is the determinant of this linear map? Consider the map $f:M_{n}(K)\to M_n(K)$
$$f(A) = A^t,A\in M_n(K).$$
What is the determinant of this map? 
After working on some examples I recognized that the matric associated with this map is a permutation matrix and so its determinant is the sign of the permutation. Excluding the matrices $E_{ii}$ we get that the number of transpositions is 
$n(n-1)/2$ thus the answer is 
$$(-1)^{n(n-1)/2}.$$ Is this answer correct?  

Yes, it is correct. Alternatively, since $f(H)=H$ for every symmetric matrix $H$ and $f(K)=-K$ for every skew-symmetric matrix $K$, the matrix space $M_n(K)$ is the sum of two eigenspaces of $f$, one of dimesion $n(n+1)/2$ for the eigenvalue $1$ and the other of dimension $n(n-1)/2$ for the eigenvalue $-1$. Hence $\det f=(-1)^{n(n-1)/2}$.
