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.}