Is A^n = I_n Enough to Prove Invertibility of A?

[auriana]

Homework Statement

Let A be an n x n matrix such that A^k = I_n for some positive integer k.
Prove that A is invertible.

Homework Equations

We have studied inverses of matrices and the Invertible Matrix Theorem, but have not yet reached determinants, just to let you know that determinants should not be used in the solution to this problem.

The Attempt at a Solution

It makes sense to me that A must be invertible in my head. I am not sure how to show this as a proof.

My first thoughts were that I could use the Theorem that states "If A and B are n x n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order," which leads to the generalization that "The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses in reverse order."

However then I realized that the theorem doesn't state that it goes backwards as well as forwards ( from AB to A and B as opposed to A and B to AB) and I wasn't sure if it was allowed.

Any help would be appreciated - thanks.

 

[Dick]

A^k=I means A*A^(k-1)=I. Think about that for a minute. What's the definition of inverse?

 

[auriana]

An nxn matrix is invertible if there is an nxn matrix C such that CA = AC = I.

So since A^k is just A*A^(k-1) or you could say its A^(k-1)*A then A^(k-1) is that C matrix and is the inverse of A and A is invertible.

Is that correct?

 

[Dick]

An nxn matrix is invertible if there is an nxn matrix C such that CA = AC = I.

So since A^k is just A*A^(k-1) or you could say its A^(k-1)*A then A^(k-1) is that C matrix and is the inverse of A and A is invertible.

Is that correct?

Sure. A^(k-1) is the inverse of A.

 

[auriana]

Thanks so much =)