# Math Help - Matrices and SLE

1. ## Matrices and SLE

Let A = [1/2 -1/2; -1/2 1/2]

Computer A^2 and A^3. What will A^n turn out to be?

I computed A^2, how can I computer A^3? Is it possible to just multiply the product of A^2 with another A? That doesn't seem right to me though. And I have no idea how to find A^n. Thanks for the help!

2. I found the answer in my book

A^3 is equal to A*A^2 and A^n is equal to [a11^(n-1) a12^(n-1); a21^(n-1) a22^(n-1)] (general form)

3. Hello,

As A^3 = A.A.A, you can choose either to calculate A².A or A.A² (the group of matrix is associative)

4. You can find $A^n$ by using diagnolization. Express $MAM^{-1} = D$ where $D$ is a diagnol matrix (which is abtained from the eigenvectors of $A$) then $A = MDM^{-1}$ and so $A^n = MD^nM^{-1}$. Since $D$ is diagnol it means the exponents $D^n$ are simply the exponents of each of its diagnol entries.