Show that $\det(A^s) = (\det(A))^s$

Question

Let A be an $n\times n$ matrix. Show that $\det(A^s) = (\det(A))^s$ for every $s\in \mathbb N = \{1,2,3,\cdots\}$

($\det$ is the determinant of the matrix).

Answer 1

Proceed by induction: For $s=1$, it's trivial. Assume true for $s=k$, prove for $s=k+1$. Then

$\det{A^{k+1}}= \det{(A^k)} det A=(\det{A})^k\det{A}=(\det{A})^{k+1}$