A) \[{{\omega }^{2}}A\]
B) \[\omega A\]
C) \[A\]
D) \[0\]
Correct Answer: B
Solution :
Given, \[A=\left[ \begin{matrix} \omega & 0 \\ 0 & \omega \\ \end{matrix} \right]\] \[{{A}^{2}}=\left[ \begin{matrix} \omega & 0 \\ 0 & \omega \\ \end{matrix} \right]\left[ \begin{matrix} \omega & 0 \\ 0 & \omega \\ \end{matrix} \right]=\left[ \begin{matrix} {{\omega }^{2}} & 0 \\ 0 & {{\omega }^{2}} \\ \end{matrix} \right]\] \[{{A}^{3}}=\left[ \begin{matrix} {{\omega }^{2}} & 0 \\ 0 & {{\omega }^{2}} \\ \end{matrix} \right]\left[ \begin{matrix} \omega & 0 \\ 0 & \omega \\ \end{matrix} \right]=\left[ \begin{matrix} {{\omega }^{3}} & 0 \\ 0 & {{\omega }^{3}} \\ \end{matrix} \right]\] Similarly, \[{{A}^{50}}=\left[ \begin{matrix} {{\omega }^{50}} & 0 \\ 0 & {{\omega }^{50}} \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} {{({{\omega }^{3}})}^{16}}{{\omega }^{2}} & 0 \\ 0 & {{({{\omega }^{3}})}^{16}}{{\omega }^{2}} \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} {{\omega }^{2}} & 0 \\ 0 & {{\omega }^{2}} \\ \end{matrix} \right]=\omega A\]
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