Matrices proof by induction

Question

For any 3x3 matrix $A$, prove by induction that $$(A^T)^n=(A^n)^T$$ for all $n∈ℕ$

I'm not sure how I do this.

Answer 1

Hint:

For $\;n=1\;$ is trivial , so assume for general $\;n\;$:

$$\left(A^t\right)^{n+1}=\left(A^t\right)^nA^t=\left(A^n\right)^tA^t$$

and now you may want to use that $\;(CD)^t=D^tC^t\;$

Answer 2

Alternatively: (But not so elegant)

$$\underbrace{(A\cdot A \dots A)^T}_{n-times}=\underbrace{(A\cdot A \dots A)^T}_{(n-1)-times}\cdot A^T=\dots=\underbrace{A^T\dots A^T}_{n-times}$$