Finding x and y using eigenvectors

[RicolaTrent]

Just wondering if you aren't given a 2x2 matrix, but call it A, but you are given an eigenvector of (5 -2) - should be in 2x1 form - and an eigenvalue of 5. How would you calculate A³ to find x and y? My thought was to cube the scalar (5) and then use the result to apply to the eigenvector, but I'm not sure if this is correct or how else to go about it!

 

[ksdhart2]

What are \(x\) and \(y\)? You've not defined these variables.

 

[Romsek]

You're going to need both eigenvalues to find the powers of A.
One eigenvalue and vector is not enough info to uniquely find the other of a given matrix.

 

[RicolaTrent]

x and y are the variables that would be in the 2x1 form of (x y). All that is supplied is the eigenvector (5 -2) and the eigenvalue of 5 to find A[³

 

[ksdhart2]

I'm still not 100% sure what you're saying, but it sounds like \(x\) and \(y\) represent the components of the other, unknown eigenvector. Is that correct? Assuming that it is, then Romsek's already answered your question in that it's not possible to find unique numerical values given only what you know.