Question

If $a,bϵR$ and
$\Delta =\left|\begin{array}{ccc}a& a+bi& bi\\ a+bi& bi& a\\ bi& a& a+bi\end{array}\right|$
then $\Delta$ equals

• A. $a^3+b^{3}i$
• B. $2(a^3+b^3)$
• C. $a^3-b^{3}i$
• D. $-2(a^3-b^3)$

Answer 1

Answer: (D)

$-2(a^3-b^3)$

$\Delta =\left|\begin{array}{ccc}a& a+bi& bi\\ a+bi& bi& a\\ bi& a& a+bi\end{array}\right|$

Applying $C_1\to C_1+C_2+C_3$

$\Delta =\left|\begin{array}{ccc}2a+2bi& a+bi& bi\\ 2a+2bi& bi& a\\ 2a+2bi& a& a+bi\end{array}\right|=(2a+2bi)\left|\begin{array}{ccc}1& a+bi& bi\\ 1& bi& a\\ 1& a& a+bi\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =(2a+2bi)\left|\begin{array}{ccc}1& a+bi& bi\\ 0& -a& a-bi\\ 0& -bi& a\end{array}\right|$

$=(2a+2bi)(-a^2+bi(a-bi))=2(a+bi)(-a^2+abi+b^2)$

$=-2(a^3-b^3)$

Hence, the option 'D' is correct.