2 quick questionon linear algebra

ok i got quick questions i have a question that asks me to prove that a complex vector times its complex conjugate transpose is real. I know that this is true just need a little help on notation, what is the proper way to define the complex vector?

and the second question what property of a matrix would give you atleast one zero eigenvalue the matrix in question looks almost diagonal...so would dominatly diagonal be the property? and then i need to prove this property.