Linear Algebra: Find A for a 2x2 matrix and when A^1001 = I

[mahrap]

1. Find A, a 2x2 matrix, where A^{1001}=I_{2}2. I know that that if A^{2}=I_{2}, then A is either a reflection or a rotation by π.
3. If I use advantage of that fact that A in A^{2}=I_{2} is a rotation by π then I know that A^{1001}=I_{2} is true when A is a rotation by 2π/1001

Is there any other way to do this problem or is it only solvable by considering the fact that A is a rotation. Can you do it by considering A to be a reflection?

 

[Dick]

1. Find A, a 2x2 matrix, where A^{1001}=I_{2}


2. I know that that if A^{2}=I_{2}, then A is either a reflection or a rotation by π.



3. If I use advantage of that fact that A in A^{2}=I_{2} is a rotation by π then I know that A^{1001}=I_{2} is true when A is a rotation by 2π/1001

Is there any other way to do this problem or is it only solvable by considering the fact that A is a rotation. Can you do it by considering A to be a reflection?

You said it yourself. A reflection satisfies A^2=I. How can one satisfy A^1001=I?