Question

The value of the determinant $\left|\begin{array}{ccc}1+i& 1-i& i\\ 1+i& i& 1+i\\ i& 1+i& 1-i\end{array}\right|$

• A. $2+5i$
• B. $7-4i$
• C. $4+7i$
• D. $-2+5i$

Answer 1

The correct option is D $-2+5i$

We have, $\left|\begin{array}{ccc}1+i& 1-i& i\\ 1+i& i& 1+i\\ i& 1+i& 1-i\end{array}\right|$
$R_1\to R_1-R_2$
$R_2\to R_2-R_3$ $\left|\begin{array}{ccc}0& 1-2i& -1\\ 1& -1& 2i\\ i& 1+i& 1-i\end{array}\right|$
$\therefore determinant=-1\left[\right(1-2i\left)\right(1-i)-(1+i\left)\right(-1\left)\right]+i\left[\right(1-2i)2i-1]$$-1-i-(1-2-3i)+i[2i+4-1]$
$=-1-i+1+3i-2+3i$
$=-2+5i$