Trying to simplify Root expressions from the output of Eigenvalues

I am trying to calculate eigenvalues of a sparse matrix with only two distinct non-zero elements, here Alpha and Beta, which are both negative reals. Mathematica returns some complex expressions with Root[] values when using the Eigenvalues[] command on the following matrixA:

In all cases the matrices are symmetric and real and hence have real eigenvalues.

matrixA={
{α, β, 0, 0, 0, 0, β, 0, 0, β},
{β, α, β, 0, 0, 0, 0, 0, 0, 0},
{0, β, α, β, 0, 0, 0, 0, 0, 0},
{0, 0, β, α, β, 0, 0, 0, 0, 0},
{0, 0, 0, β, α, β, 0, 0, 0, 0},
{0, 0, 0, 0, β, α, β, 0, 0, 0},
{β, 0, 0, 0, 0, β, α, β, 0, 0},
{0, 0, 0, 0, 0, 0, β, α, β, 0},
{0, 0, 0, 0, 0, 0, 0, β, α, β},
{β, 0, 0, 0, 0, 0, 0, 0, β, α}
}


For comparison, with all the other similar matrices I've tried (see below e.g. matrixB) Mathematica will put out simple decimal approximations (using Eigenvalues[matrixB] // N // Simplify)

Can anyone point out a way to get expressions for the matrixA as simple as for matrixB?

And yes, the desired simple answers for matrixA do exist, I can get them with other programs, but I want to use Mathematica!