Question:
If $\omega \ne 1$ is the complex cube root of unity and matrix $ H =\begin{bmatrix}\omega&0\\ 0&\omega\end{bmatrix}$, then $H^{70}$ is equal to -
If $\omega \ne 1$ is the complex cube root of unity and matrix $ H =\begin{bmatrix}\omega&0\\ 0&\omega\end{bmatrix}$, then $H^{70}$ is equal to -
Updated On: Jul 5, 2022
- 0
- -H
- $H^2$
- H
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The Correct Option is D
Solution and Explanation
$H^{2} = \begin{bmatrix}\omega&0\\ 0&\omega\end{bmatrix}\begin{bmatrix}\omega &0\\ 0&\omega \end{bmatrix} = \begin{bmatrix}\omega^{2} &0\\ 0&\omega^{2} \end{bmatrix}$
If $H^{k} = \begin{bmatrix}\omega^{k} &0\\ 0&\omega^{k} \end{bmatrix}$, then $H^{k+1} = \begin{bmatrix}\omega^{k+1} &0\\ 0&\omega^{k+1} \end{bmatrix}$
So by mathematical induction,
$ H^{70} = \begin{bmatrix}\omega ^{70} &0\\ 0&\omega ^{70} \end{bmatrix} = \begin{bmatrix}\omega &0\\ 0&\omega \end{bmatrix} = H$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
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