# Thread: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

1. ## A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

Let A be a symmetric matrix over the real numbers. A^10 = I. Prove A^2 = I.

I had this question in an exam but got 0/6 for it, so I'm not going to show my working here which was just wrong. How should I go about solving this question.

Thanks

2. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

Hint: A symmetric matrix over $\mathbb{R}$ is diagonalizable, and the eigenvalues are real. What about the eigenvalues of $A$?

3. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

Originally Posted by girdav
Hint: A symmetric matrix over $\mathbb{R}$ is diagonalizable, and the eigenvalue are real. What about the eigenvalues of $A$?
The diagonal of A is made of eigenvalues and A^10 = I, so A's eigenvalues are from the set {1,-1}. So A^2 =1.
Thanks!

But how do I know that a symmetric matrix is diagonalizable?

4. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

It's true for symmetric real matrices. It's a classical result, which can be shown for example by induction on dimension of the vector space. You can find a proof here.

5. ## Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

Ah! Thanks. That's what the real spectral theorem tells me!