 Solution

The same invariant subspaces work for $S=\alpha I+\beta T$: it acts by $\alpha$ on $\ker T$ and by $\alpha +\beta$ on $\mathrm{im}T$. Its minimal polynomial is therefore the product of the distinct linear polynomials belonging to the nonzero summands. Explicitly,When $\beta \ne 0$, the two candidate eigenvalues are distinct in any field. This is the minimal polynomial of an affine function of a projection.