If omega neq 1 is a cube root of unity and H | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $H = \begin{bmatrix} \omega & 0 \ 0 & \omega \end{bmatrix}$.We calculate the powers of $H$:$H^2 = \begin{bmatrix} \omega & 0 \ 0 & \omega

Vedclass · Accepted Answer

$H$

Vedclass · Answer

(C) Given $H = \begin{bmatrix} \omega & 0 \ 0 & \omega \end{bmatrix}$.We calculate the powers of $H$:$H^2 = \begin{bmatrix} \omega & 0 \ 0 & \omega \end{bmatrix} \begin{bmatrix} \omega & 0 \ 0 & \omega \end{bmatrix} = \begin{bmatrix} \omega^2 & 0 \ 0 & \omega^2 \end{bmatrix}$.By induction,$H^n = \begin{bmatrix} \omega^n & 0 \ 0 & \omega^n \end{bmatrix}$.For $n = 70$,$H^{70} = \begin{bmatrix} \omega^{70} & 0 \ 0 & \omega^{70} \end{bmatrix}$.Since $\omega^3 = 1$,we have $\omega^{70} = (\omega^3)^{23} \cdot \omega = (1)^{23} \cdot \omega = \omega$.Therefore,$H^{70} = \begin{bmatrix} \omega & 0 \ 0 & \omega \end{bmatrix} = H$.

If $\omega \neq 1$ is a cube root of unity and $H = \begin{bmatrix} \omega & 0 \\ 0 & \omega \end{bmatrix}$,then $H^{70}$ is equal to:

Similar Questions

Find the value of $x, y$ and $z$ from the following equation: $\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$

If $A + B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $A - 2B = \begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix}$,then $A=$

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$,then $A^{100} = $ . . . . . . .

Construct a $2 \times 2$ matrix,$A = [a_{ij}]$,whose elements are given by: $a_{ij} = \frac{(i+j)^2}{2}$

If $A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix}$ and $hA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}$, then the values of $h, a, b$ are respectively

Vedclass Test Series

Exam Paper Generator

Online Exam Module