Question

Question #47f8c

Answer 1

False

Explanation:

For a statement to be false, a single counterexample is enough. The identity matrix does satisfy `A^2=A` - but its only eigenvalues are 1. On the other hand any square null matrix satisfies this equation as well - but its only eigenvalues are 0.

Of course, if `A^2 =A` and `x` is an eigenvector of `A` belonging to eigenvalue `lambda`, then

`A^2x=Ax implies lambda^2x=lambda x implies lambda^2 = lambda`
this means that the only possible values of an eigenvalue is given by 0 or 1 - but this does not imply that both values have to be available for every matrix that obeys `A^2=A`