Give a proof if the statement is true, or give a counterexample if it is false:
a) If $A=A^*$, then $A+iI$ is invertible.
b) If $U$ is unitary, $U + 3/4 I$ is invertible.
c) If a matrix $A$ is real, $A - iI$ is invertible.
So for a), I was thinking this is true. If $A=A*$ and $A$ is $n × n$, then $A$ has $n$ real eigenvalues. So then $A+iI$ will have $n$ complex eigenvalues, and none will be zero. So then $A+iI$ will be invertible. Is this correct?
b) This one I am unsure. It makes sense when I try with matrices so I think it is true, but I am unsure how to go about proving it.
c) I think it is true. My proof would be by contradiction. So let $A-Ii$ not be invertible. So there is a solution to $(A-iI)x=0 $such that $x\neq 0$. So $Ax-ix=0$. So then $Ax=ix$. So then $i$ is an eigenvalue of $A$. But $A$ is real so it has to have real eigenvalues. Contradiction. Therefore $A-iI$ is invertible.
Please let me know if I am on the right track with these!