Question

For the matrices Gand K in$\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{21}\mathbf{)}$, find the matrices$\mathbf{R}\mathbf{=}\mathbf{GK}$ and$\mathbf{S}\mathbf{=}\mathbf{KG}$. Note that$\mathbf{R}\mathbf{\ne }\mathbf{S}$. (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short answer

The geometric transformations are that the rotation axis of R its z axis and the rotation angle is ${90}^{\circ }$and that of S is x axis and the rotation angle is ${90}^{\circ }$

Step-by-step solution

Matrix Transformations

Matrix transformation can be of two types: rotation and reflection only for square matrices. When the determinant value of the matrix is1then it is termed rotation and if the value is -1 then, it is a reflection.

Given Parameters

$\mathrm{G}=\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ 1& 0& 0\end{array}\right)\mathrm{and}\mathrm{K}=\left(\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right)$

Calculating R and S

$$\begin{gathered} \mathrm{R}=\mathrm{GK} \\ =\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right) \\ =\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right) \\ \mathrm{S}=\mathrm{KG} \\ =\left(\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ 1& 0& 0\end{array}\right) \\ =\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right) \end{gathered}$$

Therefore,$R\ne S$and G ,K do not commute.

Determinants of R and S need to be calculated.

$detR=1$and $detS=1$hence both matrices are rotations.

The rotation axis of R its z axis and the rotation angle is ${90}^{\circ }$and that of S is x axis and the rotation angle is ${90}^{\circ }$.