2014-42.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	42	IB 2014 Linear Algebra	Paper 2, Section II, G	2014

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) $\operatorname{det}\left(\begin{array}{cc}A & -B \ B & A\end{array}\right)=\operatorname{det}(A+i B) \operatorname{det}(A-i B)$.