Let \(z\) and \(w\) be arbitrary complex numbers. Put \(A=zw^*\). Use conjugate laws; do not expand \(z\) and \(w\) into Cartesian form.
Show that \(A^*=z^*w\), and hence that \(zw^*+z^*w\) is real.[3]
Show that \(zw^*-z^*w\) has zero real part. Explain why it need not be a nonzero purely imaginary number.[3]
Show that \(\dfrac{zw^*-z^*w}{i}\) is real.[2]
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Answer:(a) \(A^*=z^*w\); the sum is \(2\operatorname{Re}(A)\). (b) The difference is \(2i\operatorname{Im}(A)\) and can be zero. (c) \(2\operatorname{Im}(A)\in\mathbb R\).