1. ## Find Matrix A^n...

Hello,

I have a very small doubt on this problem..

A = (1 4 2 3) in M(2x2) R, find an expression for A^n, where n is an arbitrary positive integer.

I know its a pretty simple concept but I am getting hard time figuring it out!

when I multiply A*A I get = (9 16 8 17)

and then next will be A^2 * A = (41 84 42 83)

and then next will be A^3*A = (209 416 208 417)

and then next will be A^4*A = (1041 2084 1042 2083)

I am sure there is a pattern but I cant seem to understand and simplify this problem!

thanks a lot!

2. Google: the cayley hamilton theorem

3. There won't be any simple pattern. You need to do it a different way.

That matrix has eigenvalues -1 and 5. That means that if we define "P" to be the matrix having the corresponding eigenvectors as columns, we have
$\displaystyle \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}= P^{-1}\begin{bmatrix}-1 & 0 \\ 0 & 3\end{bmatrix}P$

Writing that as $\displaystyle A= P^{-1}DP$, $\displaystyle A^2= (P^{-1}DP)(P^{-1}DP)= P^{-1}D^2P$, $\displaystyle A^3= A(A^2)= (P^{-1}DP)(P^{-1}D^2P)= P^{-1}D^3P$, etc. And, because D is diagonal, $\displaystyle D^n$ is easy.