= Solution
A real square <orthogonal matrix> satisfies $Q^TQ=I$. A complex square <unitary matrix> satisfies $U^\dagger U=I$, where $\dagger$ denotes conjugate transpose. A <Hermitian matrix> satisfies $H^\dagger=H$. The distinction between transpose and conjugate transpose matters for imaginary entries.
For the three displayed matrices, in their source order:
* First: a <unitary matrix>, but neither an <orthogonal matrix> nor a <Hermitian matrix>.
* Second: an <orthogonal matrix>, a <unitary matrix>, and a <Hermitian matrix>.
* Third: a <Hermitian matrix>, but neither an <orthogonal matrix> nor a <unitary matrix>.
For the first matrix, $A^\dagger=-A$ and $A^2=-I$, so $A^\dagger A=I$ but $A^\dagger\ne A$. It also fails $A^TA=I$, even if complex orthogonality is considered. The second matrix is real diagonal with entries $1,-1$: its transpose, conjugate transpose and inverse all equal itself. In the third matrix the imaginary off-diagonal entries are conjugate pairs, establishing the <Hermitian matrix> condition. Its first column has squared norm $1+3=4$, so it is not a <unitary matrix>; it is not real orthogonal, and the complex bilinear first-column square is $1-3=-2\ne1$ as well.
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