Is the set of all 3 × 3 matrices that have the vector #[2, 1 , -2]^T# as an eigenvector closed under addition?
Yes, see below.
A set is closed under addition if the sum of any two elements in the set is also in the set.
Here, if 2 matrices in the set are
#M_1 mathbf e = lambda_1 mathbf e#; and
#M_2 mathbf e = lambda_2 mathbf e#
It follows from adding these together that: