Define the determinant of an $n \times n$ complex matrix $A$. Explain, with justification, how the determinant of $A$ changes when we perform row and column operations on $A$.

Let $A, B, C$ be complex $n \times n$ matrices. Prove the following statements. (i) $\operatorname{det}\left(\begin{array}{cc}A & C \\ 0 & B\end{array}\right)=\operatorname{det} A \operatorname{det} B$. (ii) $\operatorname{det}\left(\begin{array}{cc}A & -B \\ B & A\end{array}\right)=\operatorname{det}(A+i B) \operatorname{det}(A-i B)$.