Past exam of the mathematical methods I course of the University of Oxford / 2019 / 2

The determinant is $z\cdot z^3-(-i)(-i)=z^4+1$. Thus $z^4=-1=e^{i(\pi +2\pi n)}$, and its four polynomial roots areIn Cartesian form these are $(1+i)/\sqrt{2}$, $(-1+i)/\sqrt{2}$, $(-1-i)/\sqrt{2}$ and $(1-i)/\sqrt{2}$. Their moduli are one and their complex arguments are equally spaced by $\pi /2$. The Argand diagram is therefore a square centered at the origin, with sides parallel to the coordinate axes.