Showing A Matrix Property Is True

[Bashyboy]

Homework Statement

Let A be an invertible matrix. Show that $(A^n)^{-1} = (A^{-1})^n$

Homework Equations

The Attempt at a Solution

I want to begin on the left side of the equality sign; but I am having a little difficulty on expanding it. I started to--$(A^n)^{-1} = AAA...A^{-1}$--but it just didn't appear correct. Could someone help me?

 

[HallsofIvy]

I would be inclined to prove it using "induction". If n= 1, this just says that $(A)^{-1}= A^{-1}$. Now, suppose $(A^k)^{-1}= (A^{-1})^k$. What can you say about $(A^{k+1})^{-1}= ((A^k)A)^{-1}$?

(You will need to prove separately that $(AB)^{-1}= B^{-1}A^{-1}$.)

By the way, you have

want to begin on the left side of the equality sign; but I am having a little difficulty on expanding it. I started to--(Aⁿ)⁻¹ =AAA...A⁻¹ --but it just didn't appear correct. Could someone help me?

.
No, that is not correct. (Aⁿ)^-1 is the multiplicative inverse of Aⁿ: (A^{n)(An)-1= I.}