Question

Find the value of the following determinants, where $i=\sqrt{-1}.$
(i) $\left|\begin{array}{cc}2i& -3i\\ i^3& {-2i}^5\end{array}\right|$ (ii)$\left|\begin{array}{cc}1+3i& i-2\\ -i-2& 1-3i\end{array}\right|$

Answer 1

(ii) $\left|\begin{array}{cc}2i& -3i\\ i^3& -2i^5\end{array}\right|$

$=\left(2i\right)(-2i^5)-\left(i^3\right)(-3i)$

$=-4i^6+3i^4$

$=i^4(-4i^2+3)$ $[\because i^2=-1,i^4=1]$

$=-4(-1)+3$

$=4+3$

$=7$.

(ii) $\left|\begin{array}{cc}1+3i& i-2\\ -i-2& 1-3i\end{array}\right|$

$=(1+3i)(1-3i)-(-i-2)(-i-2)$

$=(1+3i)(1-3i)+(i-2)(i+2)$

$=\{1^2-(3i{)}^2\}+\{i^2-\left(2{)}^2\right\}$

$=(1-9)+(-1-4)$

$=10-5$

$=5$.