Question

Value of $\left|\begin{array}{c}3i\\ 5\\ i\end{array}\begin{array}{c}2i\\ 4\\ 2i\end{array}\begin{array}{c}2i\\ -3i\\ 7\end{array}\right|$ is

• A. $2i+12$
• B. $2i-12$
• C. $-2i-12$
• D. $-2i+12$

Answer 1

The correct option is B $2i-12$

Let $\Delta =\left|\begin{array}{ccc}3i& 2i& 2i\\ 5& 4& -3i\\ i& 2i& 7\end{array}\right|$

Expand the above determinant along $R_1$ as shown below:

$\Delta =3i[\left(4\right)\left(7\right)-\left(2i\right)(-3i)]-2i[\left(5\right)\left(7\right)-\left(i\right)(-3i)]+2i[\left(5\right)\left(2i\right)-\left(i\right)\left(4\right)]$

$\Delta =3i[28+6i^2]-2i[35+3i^2]+2i[10i-4i]$

Since $i^2=-1$ therefore,

$\Delta =3i[28-6]-2i[35-3]+2i\left[6i\right]$

$\Delta =3i\left[22\right]-2i\left[32\right]+12i^2$

$\Delta =66i-64i-12$

$\Delta =2i-12$

Hence option $B$ is correct.