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Question:
Let $n\ge2$ be an integer,
$A=\begin{pmatrix}\cos\left(2\pi/ n\right)&\sin \left(2\pi / n\right)&0\\ -\sin\left(2\pi / n\right)&\cos\left(2\pi / n\right)&0\\ 0&0&1\end{pmatrix}$
and $?$ is the identity matrix of order $3$. Then
Let $n\ge2$ be an integer,
$A=\begin{pmatrix}\cos\left(2\pi/ n\right)&\sin \left(2\pi / n\right)&0\\ -\sin\left(2\pi / n\right)&\cos\left(2\pi / n\right)&0\\ 0&0&1\end{pmatrix}$
and $?$ is the identity matrix of order $3$. Then
Updated On: Apr 27, 2024
- $A^{n}=I$ and $A^{n-1} \ne I$
- $A^{m} \ne I$ for any positive integer $m$
- $A$ is not invertible
- $A^m = 0$ for a positive integer $m$
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The Correct Option is A
Solution and Explanation
$A=\begin{bmatrix}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
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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.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.
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