I have to find the Eigenvalues of a Matrix A. This is pretty easy but the second part of the question is:

What are the corresponding eigenvalues and eigenvectors for A^{3}-A^{-1}?

I know that the eigenvalues of A^{3}are equal to the third power of the eigenvalues of A, so this is no problem.

If the Eigenvalues of A are -2 and 3 like in my case, the Eigenvalues for A^{3}are -8 and 27.

But I don't know how to deal with the subtraction of the inverse, is there any rule or do I have to take the inverse of the matrix manually and subtract it from A^{3}?

In this case I would have to do a lot of matrix multiplication which takes a lot of time...