Question:
An orthogonal matrix is
An orthogonal matrix is
Updated On: Jul 6, 2022
- $\begin{bmatrix} \cos\,\alpha & 2\,\sin\,\alpha \\[0.3em] -2\,\sin\,\alpha & \cos\,\alpha \end{bmatrix} $
- $\begin{bmatrix} \cos\,\alpha & \sin\,\alpha \\[0.3em] -\sin\,\alpha & \cos\,\alpha \end{bmatrix} $
- $\begin{bmatrix} \cos\,\alpha & \sin\,\alpha \\[0.3em] \sin\,\alpha & \cos\,\alpha \end{bmatrix} $
- $\begin{bmatrix} 1& 1\\[0.3em] 1& 1\end{bmatrix} $
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The Correct Option is B
Solution and Explanation
Let A = $\begin{bmatrix}
\cos\,\alpha & 2\,\sin\,\alpha \\[0.3em]
-2\,\sin\,\alpha & \cos\,\alpha \end{bmatrix} $ ,then
A' = $\begin{bmatrix}
\cos\,\alpha & -\sin\,\alpha \\[0.3em]
\sin\,\alpha & \cos\,\alpha \end{bmatrix}
$
$\therefore$ AA' = $\begin{bmatrix}
\cos\,\alpha & -\sin\,\alpha \\[0.3em]
-\sin\,\alpha & \cos\,\alpha \end{bmatrix} \begin{bmatrix}
\cos\,\alpha & -\sin\,\alpha \\[0.3em]
\sin\,\alpha & \cos\,\alpha \end{bmatrix}
$
= $\begin{bmatrix}
cos^2\,\alpha + sin^2 \, \alpha & -cos \, \alpha \,sin\,\alpha\,+sin\,\alpha\,cos\,\alpha \\[0.3em]
-sin\,\alpha \, cos \, \alpha + cos \, \alpha \, sin \, \alpha &sin^2 \, \alpha + cos^2 \, \alpha \end{bmatrix}
$
= $\begin{bmatrix}1&0\\ 0&1\end{bmatrix} $ = I
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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.