Determinants of sizes two and three

$\left|\begin{array}{ll}3 & 5 \\ 5 & 3\end{array}\right|$;

$\left|\begin{array}{ll}a b & a c \\ b d & c d\end{array}\right|$;

$\left|\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right|$;

$\left|\begin{array}{ll}\sin \alpha & \sin \beta \\ \cos \alpha & \cos \beta \end{array}\right|$;

$\left|\begin{array}{cc}\log_{b} a & 1 \\ 1 & \log_{a} b\end{array}\right|$;

$\left|\begin{array}{cc}\cos \alpha +i \sin \alpha & 1 \\ 1 & \cos \alpha -i \sin \alpha \end{array}\right|$;

$\left|\begin{array}{rr}a+b i & c+d i \\ -c+d i & a-b i\end{array}\right|$.

$\left|\begin{array}{lll}1 & 2 & 3 \\ 5 & 1 & 4 \\ 3 & 2 & 5\end{array}\right|$;

$\left|\begin{array}{rrr}-1 & 5 & 4 \\ 3 & -2 & 0 \\ -1 & 3 & 6\end{array}\right|$;

$\left|\begin{array}{lll}0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0\end{array}\right|$;

$\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right|$;

$\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$;

$\left|\begin{array}{ccc}0 & a & 0 \\ b & c & d \\ 0 & e & 0\end{array}\right|$;

$\left[\begin{smallmatrix} \sin \alpha & \cos \alpha & 1 \\ \sin \beta & \cos \beta & 1 \\ \sin \gamma & \cos \gamma & 1\end{smallmatrix}\right]$;

$\left|\begin{array}{ccc}1 & 0 & 1+i \\ 0 & 1 & i \\ 1-i & -i & 1\end{array}\right|$;

$\left[\begin{smallmatrix} 1 & \varepsilon & \varepsilon^{2} \\ \varepsilon^{2} & 1 & \varepsilon \\ \varepsilon & \varepsilon^{2} & 1\end{smallmatrix}\right] \quad \left(\varepsilon =-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)$;

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \varepsilon & \varepsilon^{2} \\ 1 & \varepsilon^{2} & \varepsilon \end{smallmatrix}\right] \quad \left(\varepsilon =\cos \frac{4}{3} \pi +i \sin \frac{4}{3} \pi \right)$.