Question

Determine if the statement is true or false, and justify your answer. the product of the eigenvalues (counting multiplicities) of a is equal to the constant term of the characteristic polynomial of
a.

Answer 1

True

The justification is that eigenvalues are the roots of the polynomial.
If roots are known, then polynomial can be written in factor form:
$P(x) = (x - e_1)(x-e_2)...(x-e_n)$
Thus the constant term is product of eigenvalues by nature of expansion of factored polynomial.

Answer 2

This may be true, you would have to compute a few and present argument though. it may have to do with how they factor and may even have deep origins to the Rational roots theorem but I am unsure at the moment.