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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Determinants

NDA 2020 · Mathematics · Determinants

What is the value of the determinant | array ccc i & i^2 & i^3 i^4...

What is the value of the determinant \left|ii2i3i4i6i8i9i12i15\begin{array}{ccc}i & i^2 & i^3 \\ i^4 & i^6 & i^8 \\ i^9 & i^{12} & i^{15}\end{array}$\right| where i=−1i=\sqrt{-1} ?$

Options

  1. A0
  2. B-2
  3. C4i4 i
  4. D−4i-4 i

Correct answer

D. −4i-4 i

Step-by-step solution

\left|ii2i3i4i6i8i9i12i15\begin{array}{ccc}i & i^2 & i^3 \\ i^4 & i^6 & i^8 \\ i^9 & i^{12} & i^{15}\end{array}\right|=\left|i−1−i1−11i1−i\begin{array}{ccc}i & -1 & -i \\ 1 & -1 & 1 \\ i & 1 & -i\end{array}\right|=i(i−1)−1(i−(−i))−i(1+i)=i2−i−2i−i−i2=−4i\begin{aligned} & =i(i-1)-1(i-(-i))-i(1+i) \\ & =i^2-i-2 i-i-i^2=-4 i\end{aligned}$$
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