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Question: \(\left| \begin{matrix} 1 + i & 1 - i & i \\ 1 - i & i & 1 + i \\ i & 1 + i & 1 - i \end{matrix} \ri...

∣1+i1−ii1−ii1+ii1+i1−i∣=\left| \begin{matrix} 1 + i & 1 - i & i \\ 1 - i & i & 1 + i \\ i & 1 + i & 1 - i \end{matrix} \right| =

A

−4−7i- 4 - 7i

B

4+7i4 + 7i

C

3+7i3 + 7i

D

7+4i7 + 4i

Answer

4+7i4 + 7i

Explanation

Solution

Δ=(2+i)∣11i11+2i1+i121−i∣\Delta = (2 + i)\left| \begin{matrix} 1 & 1 & i \\ 1 & 1 + 2i & 1 + i \\ 1 & 2 & 1 - i \end{matrix} \right|

= (2+i)( 2 + i ) ∣0−2i−10−1+2i2i121−i∣\left| \begin{matrix} 0 & - 2i & - 1 \\ 0 & - 1 + 2i & 2i \\ 1 & 2 & 1 - i \end{matrix} \right| by R1→R1−R2R_{1} \rightarrow R_{1} - R_{2} $$R_{2} \rightarrow R_{2} - R_{3}

= (2+i){−4i2+(−1+2i)}=(2+i)(4−1+2i)(2 + i)\{ - 4i^{2} + ( - 1 + 2i)\} = (2 + i)(4 - 1 + 2i)

= (2+i)(3+2i)=4+7i(2 + i)(3 + 2i) = 4 + 7i.