TITLE: Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?
QUESTION [8 upvotes]: How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$?  
Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?

REPLY [8 votes]: We recall that the determinant of an endomorphism $T : V \to V$ is the unique scalar $c$ such that the functored map
$$\bigwedge\nolimits^{\!k}  T : \bigwedge\nolimits^{\!k} V \to \bigwedge\nolimits^{\!k} V$$
is multiplication by $c$. Here $k$ is the dimension of $V$. Thus you are asking why if we apply $\bigwedge^k T$ $n$ number of times the resulting map is multiplication by $c^n$. But this is just obvious.