I am given a set of matrices such as $ A - tI$, where $A$ is some matrix and $t$ is a scalar and asked to find one eigenvalue and eigenvector in terms of $\lambda$ and $x$, respectively.
However, how can one do this without being given explicit matrices. I know that $(A-tI)x = \lambda x$ by definition. If I wanted to find the eigenvalue, then I would do $det(A-tI-\lambda I) =0$ and solve for $\lambda$. But again, I don't know how to solve this without knowing the matrix.
Note that $Ax=\lambda x$ if and only if $(A-tI)x=(\lambda-t)x$. That is, for any $t$ and $s$, the eigenvalues and eigenvectors of $A-sI$ are wholly determined by the eigenvalues and eigenvectors of $A-tI$.
Takeaway: you only need to compute the eigenvalues and eigenvectors for one matrix.