91Ó°ÊÓ

A plane rotation matrix R is a matrix that represents the rotation in a plane. It is an n × n matrix and can be defined as: \(R= \begin{bmatrix} c&-s&&&\\ s&c&&&\\ &&1&&\\ &&&\ddots&\\ &&&&1 \end{bmatrix} \) where c = cos(θ) and s = sin(θ) are the rotation parameters and θ is the angle of rotation.

Now, we need to find the determinant of the plane rotation matrix R. As R is an n × n matrix, we start calculating the determinant by expanding along the first row. We have: \(\operatorname{det}(R) = c\operatorname{det}(R_1 - s\operatorname{det}(R_2))\) where \(R_1\) and \(R_2\) are the submatrices obtained by removing the first row and first column & second row and first column from R, respectively. Notice that \(R_1\) and \(R_2\) are diagonal matrices, and the determinant of a diagonal matrix is the product of its diagonal elements: \(\operatorname{det}(R_1) = c_1c_2 \cdots c_{n-1}\) \(\operatorname{det}(R_2) = c_2c_3 \cdots c_{n-1}\) Now, substitute the values of \(\operatorname{det}(R_1)\) and \(\operatorname{det}(R_2)\) in the determinant expression: \(\operatorname{det}(R) = c(c_1c_2 \cdots c_{n-1} - s(c_2c_3 \cdots c_{n-1}))\) As we know that c = cos(θ) and s = sin(θ), we can simplify further: \(\operatorname{det}(R) = \cos(\theta)(\cos(\theta_1)\cos(\theta_2) \cdots \cos(\theta_{n-1}) - \sin(\theta)(\cos(\theta_2)\cos(\theta_3) \cdots \cos(\theta_{n-1})))\)

An elementary orthogonal matrix has the property that its determinant is either 1 or -1. In other words, an elementary orthogonal matrix preserves the orientation of the space. Now let's check if the determinant of R is 1 or -1: \(\operatorname{det}(R) = \cos(\theta)(\cos(\theta_1)\cos(\theta_2) \cdots \cos(\theta_{n-1}) - \sin(\theta)(\cos(\theta_2)\cos(\theta_3) \cdots \cos(\theta_{n-1})))\) Since the cosine and sine functions have values between -1 and 1, the determinant of R can have values other than 1 and -1. Therefore, R is not an elementary orthogonal matrix.