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Mathematics Question on Determinants

The complex number z=∣23+i−3 3−i0−1+i −3−1−i4∣z = \begin{vmatrix}2&3+i&-3\\\ 3-i&0&-1+i\\\ -3&-1-i&4\end{vmatrix} is equal to

A

3−4i3-4i

B

5+4i5+4i

C

−5i-5i

D

None ofthese

Answer

None ofthese

Explanation

Solution

Given that:z=∣23+i−3 3−i0−1+i −3−1−i4∣z=\left|\begin{matrix}2&3+i&-3\\\ 3-i&0&-1+i\\\ -3&-1-i&4\end{matrix}\right|
Expand w.r.t. ′R1′'R_{1}',
=2\left\\{\left(1+i\right)\left(i-1\right)\right\\}-\left(3-i\right)\left\\{4\left(3-i\right)+3\left(i-1\right)\right\\}+3\left\\{\left(1+i\right)\left(3-i\right)\right\\}
=2\left\\{\left(i^{2}-1\right)\right\\}-\left(3+i\right)\left\\{12-4i+3i-3\right\\}+3\left\\{3+3i-i-i^{2}\right\\}
=2(−1−1)−(3+i)(−i+9)+3(2i+4)=2\left(-1-1\right)-\left(3+i\right)\left(-i+9\right)+3\left(2i+4\right)
=-4-\left\\{-3i+27-i^{2}+9i\right\\}+6i+12
=−4−6i−28+6i+12=−20=-4-6i-28+6i+12=-20 (Purely Real)