Inaccurate zero eigenvalues for 7*7 matrix

I have symbolic entries for all the elements of a 7*7 matrix. At the symbolic level Eigenvalues gives two zeroes and five others that are extremely complicated. At the same time Det evaluates to exactly zero.

I take this to mean that, no matter what values the symbols within the matrix are assigned, I will have at least two zero eigenvalues. Exactly zero eigenvalues.

However, when I assign values to the symbols by hand, for eg., a=10; b=50, and so on, and evaluate the same codes, Eigenvalues evaluates to give two eigenvalues of the order of 10^-12. For my purpose, this magnitude cannot be treated as a zero. And I am also in need of a different eigenvalue of the same matrix which is very small, of this order or even smaller, but is not exactly zero. So I need the zeroes to show up much more accurately.

I have tried adding $ MinPrecision=50 to my code before executing the rest of it. It does not help at all.

A similar question was posted, Is there a good way to check, whether a small value produced numerically is a symbolic zero? However I could not decipher anything of use out of it.