JEE Advanced 2013Paper 2
- Exam
- JEE Advanced
- Year
- 2013
- Paper
- Paper 2
- Subject
- Mathematics
- Chapter
- Matrices
JEE Advanced 2013 · Paper 2 · Mathematics · Matrices
Let ω be a complex cube root of unity with ω ≠ 1 and P = [ p _ ij ]...
Let \omega \ne 1\text{P} = be a n x n matrix with . Then , when n =
Options
- A57
- B55
- C58
- D56
Correct answer
B. 55
Step-by-step solution
n = 1 n = 2
\left[{ω}^{2}\right] \text{P} = \left[\begin{cases}{ω}^{2} & {ω}^{3} \\ {ω}^{3} & {ω}^{4}\end{cases}\right] = \left[\begin{cases}{ω}^{2} & 1 \\ 1 & ω\end{cases}\right]{\text{P}}^{2} = \left[{ω}^{4}\right] ≠ 0 {\text{P}}^{2} = \left[\begin{cases}{ω}^{4} + 1 & \text{.......} \\ \text{.......} & \text{.......}\end{cases}\right] ≠ 0\text{P} = \left[\begin{cases}{ω}^{2} & 1 & ω \\ 1 & ω & {ω}^{2} \\ ω & {ω}^{2} & 1\end{cases}\right] \left[\begin{cases}{ω}^{2} & 1 & ω \\ 1 & ω & {ω}^{2} \\ ω & {ω}^{2} & 1\end{cases}\right] = \left[\right]{\text{P}}^{2} ≠ 0 when n is not multiple of 3.
\left[{ω}^{2}\right] \text{P} = \left[\begin{cases}{ω}^{2} & {ω}^{3} \\ {ω}^{3} & {ω}^{4}\end{cases}\right] = \left[\begin{cases}{ω}^{2} & 1 \\ 1 & ω\end{cases}\right]{\text{P}}^{2} = \left[{ω}^{4}\right] ≠ 0 {\text{P}}^{2} = \left[\begin{cases}{ω}^{4} + 1 & \text{.......} \\ \text{.......} & \text{.......}\end{cases}\right] ≠ 0\text{P} = \left[\begin{cases}{ω}^{2} & 1 & ω \\ 1 & ω & {ω}^{2} \\ ω & {ω}^{2} & 1\end{cases}\right] \left[\begin{cases}{ω}^{2} & 1 & ω \\ 1 & ω & {ω}^{2} \\ ω & {ω}^{2} & 1\end{cases}\right] = \left[\right]{\text{P}}^{2} ≠ 0 when n is not multiple of 3.