Show that this operator is Hermitian?

Question

The operator is defined as followed: $\hat A = a|u_{1}\rangle\langle u_{1}| + b|u_{2}\rangle\langle u_{2}|$

$a)\qquad a = b = 1$

$b)\qquad a = i, b = -i$

Answer 1

taking adjoint of $\ a|x\rangle\langle y|\ $ gives $\ a^*|y\rangle\langle x|\ $. And their adjoint of a sum (of ket-bra) is the sum of the adjoints