Is det(A^n) equal to det(A)^n for all matrices A?

[Hemmer]

Question. Prove det(A^n) = det(A) ^ n

I have:

det(AB) = det(A)det(B) (holds for any number of factors)
set A=B
det(AA) = det(A)det(A)
det(A^2) = det(A)^2 ?

Is this enough for a proof?

 

[radou]

Is this enough for a proof?

Of course not, since you didn't prove what you need to prove. Hint: induction.