Thread: 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!

Google: the cayley hamilton theorem

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


Writing that as , , , etc. And, because D is diagonal, is easy.