Question

$\left|\begin{array}{ccc}1+i& 1-i& i\\ 1-i& i& 1+i\\ i& 1+i& 1-i\end{array}\right|$ (where $i=\sqrt{-1}$ ) equals

• A. $7+4i$
• B. $7-4i$
• C. $4+7i$
• D. $4-7i$

Answer 1

The correct option is C $4+7i$

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

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

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

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

$=1+1-1+3+3i+i+3i$

$=4+7i$